Prove that if an converges to a nonzero constant then 1/an is bounded

Assume is not bounded, i.e. there are no for which for all .

Now, is to say that for any , we can find a large enough such that whenever . Simultaneously, this means that is bounded.

Let's suppose without loss of generality that for any . (Note that if for some finite number of values of , we can simply remove them from consideration.)

So we have



Because is bounded, we know there is some such that for all . Now,








But we initially assumed that is unbounded, so the above is impossible. Thus must be bounded.