Why do you need to prove it without using least upper bounds?

I suppose you could do it by showing that it is a Cauchy sequence and using the fact that Cauchy sequences converge.

Other than that, you could try looking at all bounded sequences of rational numbers that have an upper bound. If you say that two such sequences, and are "equivalent" if and only if the sequence converges to 0, then you can DEFINE the real numbers to be the equivalence classes. That way, the theorem follows immediately.

One difficulty with not showing any work at all is that we cannot know what methods you are familiar with or what definitions you are using. There are many different ways to prove any specific theorem!