Question

A sequence is bounded if and only if it is both bounded above and bounded below

Answer 1

Let $\{a_n\}_{n=1}^{\infty}$ be a bounded sequence, that is there exists a constant M>0, such that $|a_n|\leq M$ .

Then $a_n\leq M$ and, hence, this sequence is bounded above.

We have also $a_n\geq -M$ and, hence, this sequence is bounded below.

Let $\{a_n\}_{n=1}^{\infty}$ be a sequence bounded above and bounded below.

Tthat is there exist two constants M₁ and M₂ such that

$M_1\leq a_n\leq M_2$

Let $M=\max\{|M_1|, |M_2|\}$ . Then we have

$a_n\leq M_2\leq |M_2|\leq \max\{|M_1|, |M_2|\}= M$ and

$a_n\geq M_1\geq -|M_1|\geq -\max\{|M_1|, |M_2|\}= -M$

Hence, $|a_n|\leq M$ and the sequence $\{a_n\}_{n=1}^{\infty}$ is bounded.