Can someone explain to me why this is considered not bounded?
After looking up the definition. I can't seem to understand why this wouldn't be considered bounded.
If there exists a number m such that for every n we say the sequence is bounded below. The number m is sometimes called a lower bound for the sequence.
If there exists a number M such that for every n we say the sequence is bounded above. The number M is sometimes called an upper bound for the sequence.
it would be better to say it is unbounded, because there is no bound (of course, this also means there is no limit...well, i almost hate to say this, because some people allow "infinte limits" as a valid way of expressing unbounded sequences). as CaptainBlack pointed out, assuming there could BE a bound, leads to a contradiction.
the idea is, if there is a bound, this bound is some finite number we could (in theory, anyway) actually find. but if no matter what finite number we pick, we can find a term that exceeds that bound, there must not be a bound in the first place (the term number that exceeds a given bound could conceivably be, for some sequences, quite a bit larger than the bound itself. fortunately, in this example, we can use the same "N" term as the bound we try).