[Math] If a sequence is bounded, are its subsequences bounded as well

Question

Lets say our infinite sequence is $A(n)$ and we have two subsequences of $A(n)$:

1. $A(2n)$
2. $A(2n-1)$

If $A(n)$ is a bounded sequences, does this imply the two subsequences above are also bounded?

Would the same apply vice-versa, where if we know that those two subsequences are bounded, THEN the sequence $A(n)$ is also bounded?

A formal proof to answer these questions would be appreciated!

Answer 1

1. Suppose that $(A_n)$ is bounded. Then there is $c \ge 0$ such that

$|A_n| \le c$ for all $n \in \mathbb N$.

Hence

$|A_{2n}| \le c$ for all $n \in \mathbb N$

and

$|A_{2n-1}| \le c$ for all $n \in \mathbb N$.

This shows that $(A_{2n})$ and $(A_{2n-1})$ are bounded.

2. Now suppose that $(A_{2n})$ and $(A_{2n-1})$ are bounded. Then there are $c_1, c_2 \ge 0$ such that

$|A_{2n}| \le c_1$ for all $n \in \mathbb N$

and

$|A_{2n-1}| \le c_2$ for all $n \in \mathbb N$.

Let $c:=\max\{c_1,c_2\}$. Then we have

$|A_n| \le c$ for all $n \in \mathbb N$.

Therefore $(A_n)$ is bounded.