1. (a) Define what it means to say that a sequence (an)converges to a limit l.
(b) Let (an)be a convergent sequence and (bn)be a bounded sequence. Prove, directly
from your definition, that if lim
n→∞
an= 0, then lim
n→∞
anb2
n= 0.
(c) State the Bolzano-Weierstrass theorem.
(d) Define what it means for a function f: [a, b]→Rto be continuous at a point
c∈(a, b)and what it means for a function fto be continuous on the interval
[a, b]⊂R.
(e) Using the Bolzano-Weierstrass theorem, or otherwise, prove that if f: [a, b]→Ris
a continuous function, then it attains its least upper bound on [a, b]. In other words,
there exists c∈[a, b]such that
f(c) = lub{f(x)|x∈[a, b]}.
[You can use without proof, the fact that since fis continuous on [a, b], it is bounded
on [a, b].]
2. (a) State the Monotonic Sequence Theorem.
(b) Show that
1 + 1
nn
= 2 + 1
2! 1−1
n+1
3! 1−1
n1−2
n
+... +1
n!1−1
n1−2
n... 1−n−1
n
for all n≥1.
(c) Using (b), or otherwise, show that 1 + 1
nn
< n for all n≥3.
(d) Let (an)be the sequence defined as an=n1
nfor all n≥1. Using (c), or otherwise,
show that an+1 < anfor all n≥3.
(e) Prove that (an)is a convergent sequence.
(f) Prove that a2n= 2 1
2n√anfor all n≥1and find the limit of (an).
[If required, you can use without proof the limit lim
n→∞
21
2n= 1 and the fact that the
square root function is continuous.]
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