The answer depends on the level at which you're studying the problem. I'll give hints assuming you're taking a fairly high-level course, but it may or may not be helpful to you:
There's a fairly straightforward path to a proof that you can get by combining these two ideas:
1. Represent x by its Dedekind cut, and consider the set of integers within that cut.
2. Use the fact that any nonempty set of natural numbers has a least element. This is logically equivalent to good old induction.
There's a little bit of kludging from the fact that x could be negative, but that's not too hard to patch up. The rest is just details.
If you're taking a more introductory-level class, I honestly don't know how you'd prove it without begging the question.
There's a fairly straightforward path to a proof that you can get by combining these two ideas:
1. Represent x by its Dedekind cut, and consider the set of integers within that cut.
2. Use the fact that any nonempty set of natural numbers has a least element. This is logically equivalent to good old induction.
There's a little bit of kludging from the fact that x could be negative, but that's not too hard to patch up. The rest is just details.
If you're taking a more introductory-level class, I honestly don't know how you'd prove it without begging the question.