Question

4. True or False? a) A sequence (an) converges to L if and only if, for every k ? N, there exists M ? N such that if n ? M, then |an ? L| < 1/k. b) Suppose (an) is a sequence such that for every ? > 0, there exists an interval of length ? that contains infinitely many terms of (an). Then (an) is Cauchy. c) If (an) is Cauchy, then for every ? > 0, there exists an interval of length ? that contains infinitely many terms of (an). d) Suppose (an) is Cauchy and that for every n ? N, the interval (?1/n,1/n) contains infinitely many terms of (an). Then (an) converges to 0. e) There exists a sequence (an) such that for every k ? N and every ? > 0, the interval (k ? ?, k + ?) contains infinitely many terms of (an). True or False? You do not need to prove your answers. f) There exists a sequence whose range (the set of values of its terms) is open. g) If A is a set of real numbers that is not open, then A is closed. h) Every infinite set is unbounded. i) If f is continuous on [a, b], then f is uniformly continuous on (a, b). j) Every bounded continuous function attains its maximum and minimum values.

Name: 4 true or false a a sequence an converges to l if and only if for every k n there exists m n such that if n m then an l 1k b suppose an is a sequence such that for every 0 there exists an interval of
Uploaded: 2023-04-27T23:10:17-08:00
Duration: 2 min 10 s
Channel: Numerade
Description: 4 true or false a a sequence an converges to l if and only if for every k n there exists m n such that if n m then an l 1k b suppose an is a sequence such that for every 0 there exists an interval of

          4. True or False? 
a) A sequence (an) converges to L if and only if,
for every k ? N, there exists
M ? N such that if n ? M,
then |an ? L| < 1/k.
b) Suppose (an) is a sequence such that for every ?
> 0, there exists an interval of length ? that
contains infinitely many terms of (an). Then (an) is Cauchy.
c) If (an) is Cauchy, then for every ? > 0,
there exists an interval of length ? that contains
infinitely many terms of (an).
d) Suppose (an) is Cauchy and that for
every n ? N, the interval (?1/n,1/n) contains
infinitely many terms of (an). Then (an) converges to 0.
e) There exists a sequence (an) such that for
every k ? N and every ? > 0, the
interval (k ? ?, k + ?) contains infinitely
many terms of (an).
True or False? You do not need to prove your answers.
f) There exists a sequence whose range (the set of values of
its terms) is open.
g) If A is a set of real numbers that is not
open, then A is closed.
h) Every infinite set is unbounded.
i) If f is continuous on [a, b],
then f is uniformly continuous on (a, b).
j) Every bounded continuous function attains its maximum
and minimum values.

Added by Mario M.

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