Is this a sufficient proof that there exist unique numbers n and such that x = n + , where n is an integer and $\epsilon$ ∈ [0, 1)?

Question

I am trying to create a proof for the aforementioned theorem; it seems rather obvious that it is a true statement, but I am still new to creating and rigorizing proofs, so I want to make sure I am at least on the right track.

Here is the proof that I came up with:

1. any real number either is an integer or falls between two integers, $n$ and $n + 1$
   
2. $x$ is a real number, therefore $n \le x \lt n + 1$
3. $0 \le \epsilon \lt 1$
4. add $n$ to all sides: $n \le n + \epsilon \lt n + 1$
5. because $n + \epsilon$ can be any value between $n$ and $n + 1$, and $x$ falls between $n$ and $n + 1$, it follows that any $x$ must be able to be represented as $n + \epsilon$

Although this seems (at least almost) sufficient to prove existence, I am really unsure about how to prove uniqueness in this case. I am also not sure whether it is rigorous enough. I would love your guys' input/suggestions on how to improve my proof. :)