5. True or False? You do not need to prove your answers (but know how you would prove them!). a) If (an) and (bn) are sequences such that (an) and (an + bn) both converge, then (bn) converges. b) If (an) does not tend to infinity but its range is not bounded above, then (an) has an increasing subsequence. c) Suppose the interval (3,3.99999) contains infinitely many terms of (an). Then (an) cannot converge to 4.
5. True or False? You do not need to prove your answers (but know how you would prove them!).
a) If (an) and (bn) are sequences such that (an) and (an + bn) both converge, then (bn) converges.
b) If (an) does not tend to infinity but its range is not bounded above, then (an) has an increasing subsequence.
c) Suppose the interval (3,3.99999) contains infinitely many terms of (an). Then (an) cannot converge to 4.
d) Suppose that for every ε > 0, the interval (4 − ε, 4 + ε) contains infinitely many terms of (an). Then (an) converges to 4.
e) If (an) is Cauchy and p is a number such that for all k ∈ N, |ak − p| < 1/2, then (an) converges to a limit L that lies in the interval [p − 1/2, p + 1/2].
f) If (an) is Cauchy and p is a number such that for infinitely many values of k, |ak − p| < 1/2, then (an) converges to a limit L that lies in the interval [p − 1/2, p + 1/2].
g) Suppose (an) has limit 4 and (bn) has limit L and that bn > an for infinitely many values of n. Then L > 4.
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