I was solving the statement:
"There are integers m and n such that m > 1 and n > 1 and 1/m + 1/n is an integer."
First, I tried different integers, where n=m. I tried 2,3,4... and so on, and the statement is true in every case.
proof:
Suppose m and n are integers where m > 1 and n > 1.
Let m = 2 and n = 2. We must show that 1/m + 1/n is an integer.
Then, 1/m + 1/n = 1/2 + 1/2 = 2/2 = 1.
Therefore, when m > 1 and n > 1 then 1/m + 1/n is an integer.
My problem is that I am unsure whether or not some definitions are missing in this proof.
I would appreciate some feedback.
Your proof looks fine. You demonstrated why the statement you were interested in was true by giving providing a specific example. One minor nitpick is that your first step really isn't needed, and your first and second steps sort of contradict one another. In the first step, you pick an arbitrary $n$ and $m$, but in the second, you pick a specific $n$ and $m$. But we don't want to pick an arbitrary $m > 1$ and $n > 1$. We want to pick a specfic $m$ and $n$.
Also in general, proofs aren't often written with explicit steps laid out, since this would be cumbersome. It's fine to write the proof in a less structured manner.
But for the most part, I wouldn't worry too much about using a bunch of symbolism, either. Symbols are important in math, and they allow us to write complicated ideas succinctly and unambiguously, but I'd argue it's more important to understand the concepts behind the symbols. And it looks like you've done that. I'd rather read a correct proof that's a little verbose than one that uses a lot of symbolism but is largely unintelligible.
But if you're curious, here's how I might re-write your proof to be a little shorter and more formal:
Let $m = n = 2$. Note $m > 1$ and $n > 1$, and $\frac{1}{m} + \frac{1}{n} = \frac{1}{2} + \frac{1}{2} = 1 \in \mathbb{Z}$. Thus, the statement is true.