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Theorem | caucvgprprlemdisj 7701* | Lemma for caucvgprpr 7711. The putative limit is disjoint. (Contributed by Jim Kingdon, 21-Dec-2020.) |
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Theorem | caucvgprprlemloc 7702* | Lemma for caucvgprpr 7711. The putative limit is located. (Contributed by Jim Kingdon, 21-Dec-2020.) |
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Theorem | caucvgprprlemcl 7703* | Lemma for caucvgprpr 7711. The putative limit is a positive real. (Contributed by Jim Kingdon, 21-Nov-2020.) |
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Theorem | caucvgprprlemclphr 7704* |
Lemma for caucvgprpr 7711. The putative limit is a positive real.
Like caucvgprprlemcl 7703 but without a disjoint variable
condition
between ![]() ![]() |
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Theorem | caucvgprprlemexbt 7705* | Lemma for caucvgprpr 7711. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 16-Jun-2021.) |
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Theorem | caucvgprprlemexb 7706* | Lemma for caucvgprpr 7711. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 15-Jun-2021.) |
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Theorem | caucvgprprlemaddq 7707* | Lemma for caucvgprpr 7711. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 5-Jun-2021.) |
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Theorem | caucvgprprlem1 7708* | Lemma for caucvgprpr 7711. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.) |
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Theorem | caucvgprprlem2 7709* | Lemma for caucvgprpr 7711. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.) |
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Theorem | caucvgprprlemlim 7710* | Lemma for caucvgprpr 7711. The putative limit is a limit. (Contributed by Jim Kingdon, 21-Nov-2020.) |
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Theorem | caucvgprpr 7711* |
A Cauchy sequence of positive reals with a modulus of convergence
converges to a positive real. This is basically Corollary 11.2.13 of
[HoTT], p. (varies) (one key difference
being that this is for
positive reals rather than signed reals). Also, the HoTT book theorem
has a modulus of convergence (that is, a rate of convergence)
specified by (11.2.9) in HoTT whereas this theorem fixes the rate of
convergence to say that all terms after the nth term must be within
![]() ![]() ![]() ![]() This is similar to caucvgpr 7681 except that values of the sequence are positive reals rather than positive fractions. Reading that proof first (or cauappcvgpr 7661) might help in understanding this one, as they are slightly simpler but similarly structured. (Contributed by Jim Kingdon, 14-Nov-2020.) |
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Theorem | suplocexprlemell 7712* | Lemma for suplocexpr 7724. Membership in the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.) |
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Theorem | suplocexprlem2b 7713 | Lemma for suplocexpr 7724. Expression for the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.) |
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Theorem | suplocexprlemss 7714* |
Lemma for suplocexpr 7724. ![]() |
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Theorem | suplocexprlemml 7715* | Lemma for suplocexpr 7724. The lower cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.) |
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Theorem | suplocexprlemrl 7716* | Lemma for suplocexpr 7724. The lower cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.) |
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Theorem | suplocexprlemmu 7717* | Lemma for suplocexpr 7724. The upper cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.) |
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Theorem | suplocexprlemru 7718* | Lemma for suplocexpr 7724. The upper cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.) |
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Theorem | suplocexprlemdisj 7719* | Lemma for suplocexpr 7724. The putative supremum is disjoint. (Contributed by Jim Kingdon, 9-Jan-2024.) |
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Theorem | suplocexprlemloc 7720* | Lemma for suplocexpr 7724. The putative supremum is located. (Contributed by Jim Kingdon, 9-Jan-2024.) |
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Theorem | suplocexprlemex 7721* | Lemma for suplocexpr 7724. The putative supremum is a positive real. (Contributed by Jim Kingdon, 7-Jan-2024.) |
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Theorem | suplocexprlemub 7722* | Lemma for suplocexpr 7724. The putative supremum is an upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.) |
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Theorem | suplocexprlemlub 7723* | Lemma for suplocexpr 7724. The putative supremum is a least upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.) |
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Theorem | suplocexpr 7724* | An inhabited, bounded-above, located set of positive reals has a supremum. (Contributed by Jim Kingdon, 7-Jan-2024.) |
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Definition | df-enr 7725* | Define equivalence relation for signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.) |
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Definition | df-nr 7726 | Define class of signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.) |
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Definition | df-plr 7727* | Define addition on signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.) |
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Definition | df-mr 7728* | Define multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.) |
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Definition | df-ltr 7729* | Define ordering relation on signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.4 of [Gleason] p. 127. (Contributed by NM, 14-Feb-1996.) |
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Definition | df-0r 7730 | Define signed real constant 0. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 9-Aug-1995.) |
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Definition | df-1r 7731 | Define signed real constant 1. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 9-Aug-1995.) |
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Definition | df-m1r 7732 | Define signed real constant -1. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 9-Aug-1995.) |
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Theorem | enrbreq 7733 | Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) |
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Theorem | enrer 7734 | The equivalence relation for signed reals is an equivalence relation. Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) |
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Theorem | enreceq 7735 | Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.) |
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Theorem | enrex 7736 | The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) |
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Theorem | ltrelsr 7737 | Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.) |
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Theorem | addcmpblnr 7738 | Lemma showing compatibility of addition. (Contributed by NM, 3-Sep-1995.) |
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Theorem | mulcmpblnrlemg 7739 | Lemma used in lemma showing compatibility of multiplication. (Contributed by Jim Kingdon, 1-Jan-2020.) |
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Theorem | mulcmpblnr 7740 | Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.) |
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Theorem | prsrlem1 7741* | Decomposing signed reals into positive reals. Lemma for addsrpr 7744 and mulsrpr 7745. (Contributed by Jim Kingdon, 30-Dec-2019.) |
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Theorem | addsrmo 7742* | There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.) |
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Theorem | mulsrmo 7743* | There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.) |
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Theorem | addsrpr 7744 | Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
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Theorem | mulsrpr 7745 | Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
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Theorem | ltsrprg 7746 | Ordering of signed reals in terms of positive reals. (Contributed by Jim Kingdon, 2-Jan-2019.) |
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Theorem | gt0srpr 7747 | Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.) |
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Theorem | 0nsr 7748 | The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) |
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Theorem | 0r 7749 |
The constant ![]() |
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Theorem | 1sr 7750 |
The constant ![]() |
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Theorem | m1r 7751 |
The constant ![]() |
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Theorem | addclsr 7752 | Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) |
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Theorem | mulclsr 7753 | Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) |
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Theorem | addcomsrg 7754 | Addition of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
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Theorem | addasssrg 7755 | Addition of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
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Theorem | mulcomsrg 7756 | Multiplication of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
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Theorem | mulasssrg 7757 | Multiplication of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
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Theorem | distrsrg 7758 | Multiplication of signed reals is distributive. (Contributed by Jim Kingdon, 4-Jan-2020.) |
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Theorem | m1p1sr 7759 | Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) |
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Theorem | m1m1sr 7760 | Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.) |
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Theorem | lttrsr 7761* | Signed real 'less than' is a transitive relation. (Contributed by Jim Kingdon, 4-Jan-2019.) |
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Theorem | ltposr 7762 | Signed real 'less than' is a partial order. (Contributed by Jim Kingdon, 4-Jan-2019.) |
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Theorem | ltsosr 7763 | Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) |
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Theorem | 0lt1sr 7764 | 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) |
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Theorem | 1ne0sr 7765 | 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) |
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Theorem | 0idsr 7766 | The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.) |
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Theorem | 1idsr 7767 | 1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.) |
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Theorem | 00sr 7768 | A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) |
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Theorem | ltasrg 7769 | Ordering property of addition. (Contributed by NM, 10-May-1996.) |
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Theorem | pn0sr 7770 | A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) |
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Theorem | negexsr 7771* | Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.) |
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Theorem | recexgt0sr 7772* | The reciprocal of a positive signed real exists and is positive. (Contributed by Jim Kingdon, 6-Feb-2020.) |
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Theorem | recexsrlem 7773* | The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) |
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Theorem | addgt0sr 7774 | The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.) |
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Theorem | ltadd1sr 7775 | Adding one to a signed real yields a larger signed real. (Contributed by Jim Kingdon, 7-Jul-2021.) |
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Theorem | ltm1sr 7776 | Adding minus one to a signed real yields a smaller signed real. (Contributed by Jim Kingdon, 21-Jan-2024.) |
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Theorem | mulgt0sr 7777 | The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.) |
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Theorem | aptisr 7778 | Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.) |
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Theorem | mulextsr1lem 7779 | Lemma for mulextsr1 7780. (Contributed by Jim Kingdon, 17-Feb-2020.) |
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Theorem | mulextsr1 7780 | Strong extensionality of multiplication of signed reals. (Contributed by Jim Kingdon, 18-Feb-2020.) |
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Theorem | archsr 7781* |
For any signed real, there is an integer that is greater than it. This
is also known as the "archimedean property". The expression
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Theorem | srpospr 7782* | Mapping from a signed real greater than zero to a positive real. (Contributed by Jim Kingdon, 25-Jun-2021.) |
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Theorem | prsrcl 7783 | Mapping from a positive real to a signed real. (Contributed by Jim Kingdon, 25-Jun-2021.) |
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Theorem | prsrpos 7784 | Mapping from a positive real to a signed real yields a result greater than zero. (Contributed by Jim Kingdon, 25-Jun-2021.) |
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Theorem | prsradd 7785 | Mapping from positive real addition to signed real addition. (Contributed by Jim Kingdon, 29-Jun-2021.) |
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Theorem | prsrlt 7786 | Mapping from positive real ordering to signed real ordering. (Contributed by Jim Kingdon, 29-Jun-2021.) |
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Theorem | prsrriota 7787* | Mapping a restricted iota from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.) |
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Theorem | caucvgsrlemcl 7788* | Lemma for caucvgsr 7801. Terms of the sequence from caucvgsrlemgt1 7794 can be mapped to positive reals. (Contributed by Jim Kingdon, 2-Jul-2021.) |
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Theorem | caucvgsrlemasr 7789* | Lemma for caucvgsr 7801. The lower bound is a signed real. (Contributed by Jim Kingdon, 4-Jul-2021.) |
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Theorem | caucvgsrlemfv 7790* | Lemma for caucvgsr 7801. Coercing sequence value from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.) |
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Theorem | caucvgsrlemf 7791* | Lemma for caucvgsr 7801. Defining the sequence in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.) |
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Theorem | caucvgsrlemcau 7792* | Lemma for caucvgsr 7801. Defining the Cauchy condition in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.) |
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Theorem | caucvgsrlembound 7793* | Lemma for caucvgsr 7801. Defining the boundedness condition in terms of positive reals. (Contributed by Jim Kingdon, 25-Jun-2021.) |
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Theorem | caucvgsrlemgt1 7794* | Lemma for caucvgsr 7801. A Cauchy sequence whose terms are greater than one converges. (Contributed by Jim Kingdon, 22-Jun-2021.) |
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Theorem | caucvgsrlemoffval 7795* | Lemma for caucvgsr 7801. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
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Theorem | caucvgsrlemofff 7796* | Lemma for caucvgsr 7801. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
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Theorem | caucvgsrlemoffcau 7797* | Lemma for caucvgsr 7801. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
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Theorem | caucvgsrlemoffgt1 7798* | Lemma for caucvgsr 7801. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | caucvgsrlemoffres 7799* | Lemma for caucvgsr 7801. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | caucvgsrlembnd 7800* | Lemma for caucvgsr 7801. A Cauchy sequence with a lower bound converges. (Contributed by Jim Kingdon, 19-Jun-2021.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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