Analysis 1A exercise sheet 5: Sequences divergent to infinity
and monotone sequences
1. Show that the following sequences are divergent to +∞:
(a) (an )n∈N where an =
(b) (an )n∈N where an =
n2 +1
n+3
√
for all n ∈ N.
n for all n ∈ N.
(c) (an )n∈N where an = (n + 1)2 − n2 for all n ∈ N.
2. (a) Show that if b ∈ (0, ∞), n ∈ N and n ≥ 2 then (1+b)n ≥
n(n−1)b2
.
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(b) Show that for all b ∈ (0, ∞), lim (1 + b)n − n = ∞.
n→∞
1
n
(c) Show that limn→∞ n = 1.
3. Let (an )n∈N be a sequence of real numbers such that limn→∞ an = ∞.
Show that if c ∈ (0, ∞) then limn→∞ can = ∞ and limn→∞ − − can =
−∞.
4. Let b ∈ R, (an )n∈N be a sequence of real numbers such that lim an =
n→∞
∞ and (bn )n∈N be a sequence of real numbers such that lim bn = b.
n→∞
Show that
(a) If b > 0 then lim bn an = ∞,
n→∞
(b) If b < 0 then lim bn an = −∞,
n→∞
(c) For b = 0 give examples of sequences (bn )n∈N and (an )n∈N where:
i. lim bn an = ∞,
n→∞
ii. lim bn an = a for some a ∈ R,
n→∞
iii. lim bn an = −∞.
n→∞
5. Suppose that (an ) and (bn ) are sequences where limn→∞ an = ∞ and
there exists N, k ∈ N such that bn+k ≥ an for all n ∈ N where n ≥ N .
Show that limn→∞ bn = ∞.
6. Complete the proof of the monotone convergence theorem by showing
that if (an )n∈N is monotone decreasing and bounded below then it
is convergent. (Rather than try and mimic the case for monotone
increasing sequences, consider the sequence with terms −an .)
7. Let (an )n∈N be a bounded sequence and define (bn )n∈N by
bn = inf{ak : k ≥ n}.
Show that (bn )n∈N is convergent.
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8. Show that if (an )n∈N is monotone increasing and unbounded then
limn→∞ an = ∞.
9. Let (an )n∈N be the sequence where a1 = 2 and for all n ∈ N, an+1 =
5an +2
2an +1 .
√
(a) Show that 0 ≤ an ≤ 1 + 2 for all n ∈ N.
(b) Show that (an )n∈N is convergent.
(c) Find lim an .
n→∞
10. Let (an )n∈N and (bn )n∈N be sequences such that an < bn and (an+1 , bn+1 ) ⊆
(an , bn ) for all n ∈ N. Show that both (an ) and (bn ) are convergent.
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