Mathematical Analysis

Examples Sheet 7

1. Define a sequence (an )n∈N+ recursively by:
a1 = 1,

an+1 := 14 an +

√
an .

(a) Prove that an > 1 ∀ n ∈ N+ .
(b) Prove that (an )n∈N+ is a contracting sequence.
(c) Find the value of limn→∞ an .
2. Let (an )n∈N+ be an increasing sequence which is bounded above and define A := {an : n ∈ N+ }. Prove
that limn→∞ an = sup A.
3. Let X be a nonempty subset of R which is bounded below and suppose that inf X ∈
/ X. Prove that
X contains a strictly decreasing sequence which converges to inf X.
4. Let A and B be subsets of R. Define A − B := {a − b : a ∈ A, b ∈ B}.
(a) Suppose that both A and B are bounded below. Prove that inf A ∪ B = min{inf A, inf B}.
(b) Suppose that A is bounded below and B is bounded above. Prove that inf(A−B) = inf A−sup B.
5. Find sup A and inf A where A is the set defined by
(a) A = {x ∈ Q : x|x| 6 4}.
(b) A = {x ∈ Z : x2 + x < 10}.


x
(c) A =
:x∈R .
1 + 4×2
(d) A = {x ∈ Q : xn > 0 for some n ∈ N+ }.
6. Provide an example of a subset X of R for which sup X = inf X.