mmtheorems72 - Metamath Proof Explorer

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**Statement List for Metamath Proof Explorer -** 7101-7200 - Page 72 of 108
Type	Label	Description
Statement

Theorem	climconst3 7101	A constant sequence converges to its value.
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Theorem	clim0 7102	The zero sequence converges to zero.


Theorem	climunii 7103	An infinite sequence of complex numbers converges to at most one limit.
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Theorem	climuni 7104	An infinite sequence of complex numbers converges to at most one limit.
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Theorem	climeu 7105	An infinite sequence of complex numbers converges to at most one limit.


Theorem	climreu 7106	An infinite sequence of complex numbers converges to at most one limit.


Theorem	2climnn 7107	If two sequences converge to each other, they converge to the same limit.
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Theorem	2climnn0 7108	If two sequences converge to each other, they converge to the same limit.
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Theorem	climshft 7109	A shifted function converges iff the original function converges.


Theorem	climres 7110	A function restricted to upper integers converges iff the original function converges.


Theorem	climshft2 7111	A shifted function converges iff the original function converges. (Contributed by Paul Chapman, 19-Nov-2007.)
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Theorem	iserzshft2 7112	Index shift of an infinite series. (Contributed by Paul Chapman, 19-Nov-2007.)
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Theorem	climuz0 7113	A zero sequence converges to zero.


Theorem	serzclim0 7114	The zero series converges to zero. (Contributed by Paul Chapman, 9-Feb-2007.)


Theorem	climrecl 7115	The limit of a convergent real sequence is real. Corollary 12-2.5 of [Gleason] p. 172.
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Theorem	climfnrcl 7116	The limit of a convergent real sequence on natural numbers is real. Corollary 12-2.5 of [Gleason] p. 172.


Theorem	climge0 7117	A nonnegative sequence converges to a nonnegative number.
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Theorem	climabs0 7118	Convergence to zero of the absolute value is equivalent to convergence to zero.


Theorem	climaddlem1 7119	Lemma for climadd 7122.

Theorem	climaddlem2 7120	Lemma for climadd 7122.

Theorem	climaddlem3 7121	Lemma for climadd 7122. Warning: The HTML proof page is 3/4 megabyte in size.

Theorem	climadd 7122	Limit of the sum of two converging sequences. Proposition 12-2.1(a) of [Gleason] p. 168.
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Theorem	climaddc1 7123	Limit of a constant added to each term of a sequence.
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Theorem	climaddc2 7124	Limit of a constant added to each term of a sequence.
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Theorem	climmullem1 7125	Lemma for climmul 7133.

Theorem	climmullem2 7126	Lemma for climmul 7133.

Theorem	climmullem3 7127	Lemma for climmul 7133.

Theorem	climmullem4 7128	Lemma for climmul 7133.

Theorem	climmullem5 7129	Lemma for climmul 7133. Instead of the infimum that Gleason uses (bottom of p. 170), we use recrecltt 5905 to give us a number smaller than both a given number and 1. Warning: The HTML proof page is 1/2 megabyte in size.

Theorem	climmullem6 7130	Lemma for climmul 7133.

Theorem	climmullem7 7131	Lemma for climmul 7133.

Theorem	climmullem8 7132	Lemma for climmul 7133. Warning: The HTML proof page is 3/4 megabyte in size.

Theorem	climmul 7133	Limit of the product of two converging sequences. Proposition 12-2.1(c) of [Gleason] p. 168.
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Theorem	climmulc2 7134	Limit of a sequence multiplied by a constant . Corollary 12-2.2 of [Gleason] p. 171.
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Theorem	climsub 7135	Limit of the difference of two converging sequences. Proposition 12-2.1(b) of [Gleason] p. 168.
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Theorem	climsubc2 7136	Limit of a constant minus each term of a sequence.
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Theorem	climaddc 7137	Limit of a constant added to a sequence.
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Theorem	climmulc 7138	Limit of a sequence multiplied by a constant . Corollary 12-2.2 of [Gleason] p. 171.
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Theorem	clim2serzt 7139	The limit of an infinite series with an initial segment removed. (Contributed by Paul Chapman, 9-Feb-2007.)
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Theorem	iserzext 7140	If an infinite series converges, so does the series with initial terms removed. (Contributed by Paul Chapman, 9-Feb-2007.)
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Theorem