Prove that if a sequence is eventually bounded then it is bounded

Question

Definitions:

1. eventually: A sequence $(a_n)$ satisfies a certain property eventually if there is a natural number $N$ such that the sequence $(a_{N+n})$ satisfies that property.

2. bounded: if it is both bounded above and bounded below.

3. bounded above: if there exists $U$ such that, for all $n, a_n \leq U; U$ is an upper bounded for $(a_n)$

4. bounded below: if there exists $L$ such that, for all $n, a_n \geq L;L$ is a lower bound for $(a_n)$

So I know the fact that each finite set has a maximum and a minimum; is this what I want to use in my proof?

Answer 1

Hint: Since after $N$ terms the sequence is bounded consider the finite set $\{a_1,a_2, \dots , a_N\}$ and then the sequence $a_{N+i}$ which you know is bounded. Go from there