Question

Show that the sequence (a_n) is bounded iff |a_n| is bounded .

Answer 1

ANSWER

A sequence $\left \{a _{n} \right \}$ is called bounded if there are real numbers $l$ and $b$ , such that

$l\leq a_{n}\leq b$ for all $n\in N$ . (1)

1) Let $\left \{a _{n} \right \}$ be a bounded sequence, denote $M=max\left \{ |l |, \: |b|\right \}$

Since $|b|\leq M, |l|\leq M$ , then $-M\leq l\leq M, -M\leq b\leq M$ . Hence

$-M\leq l\leq a_{n}\leq b\leq M$ for all $n\in N$ .

It means, that

$0\leq |a_{n}|\leq M$ (2)

Those sequence $\left \{|a _{n}| \right \}$ is bounded.

2) Conversely , let the sequence $\left \{|a _{n}| \right \}$ is bounded. Then exists $M>0$ such that inequality (2)

holds for all $n\in N$ ..Since (2) is equivalent to the inequality

$-M\leq a_{n}\leq M$ ,

then (1) is satisfied for $l=-M, b=M$ . Hence, the sequence $\left \{a _{n} \right \}$ is bounded.