4. True or False? You do not need to prove your answers. 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 E > 0, there exists an interval of length E that contains infinitely many terms of (an). Then (an) is Cauchy. c) If (an) is Cauchy, then for every E > 0, there exists an interval of length E 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 E > 0, the interval (k − E, k + E) contains infinitely many terms of (an).
4. True or False? You do not need to prove your answers.
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 E > 0, there exists an interval of length E that contains infinitely many terms of (an). Then (an) is Cauchy.
c) If (an) is Cauchy, then for every E > 0, there exists an interval of length E
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 E > 0, the interval (k − E, k + E) contains infinitely many terms of (an).
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