Everyone has their own way of proving things, and that's OK. The statement you made can be shown to be true in different ways, and what counts is that you prove it, not how you prove it. The way you learn how to prove it is that you prove many many other statements as well, and then you get used to it. Repetitio mater studiorum est.
However, if you want a path that leads to the particular proof, in this case, my thoughts would proceed as follows:
- I look at the statement. OK, it's saying that I can squeeze any positive real $x$ between two numbers. OK, let's imagine the $x$ as some point on the positive real line.
- Hmm, the two numbers, $n_0(n_0+1)$ and $(n_0+1)(n_0+2)$ are both integers.
- Not only are they integers, they are two integers from a monotonically increasing sequence of integers, $a_n = n(n+1)$.
- So... this sequence of integers, it's really a series of points on the real line. Let's imagine them as crosses. (yes, really, I do that. Don't judge) The order I draw them in is left to right.
- So what I now have is the statement that there exist two crosses so that the previous cross is to the left of $x$, and the next one is to the right. Well... sure they do! I just need to find the last cross on the left of $x$, and the next one must be on the right of it (otherwise, the previous one wasn't the last one!).
- OK, what I just said in point 5 can be said formally as "I need to find the maximum value of $n$ such that $a_n < x$. This can be rewritten formally as finding a maximum element from some set.
Once this train of thought concludes, I go down to really writing down the proof, and the proof "starts" with introducing the set. Sure, the proof does, but the thought process that lead me to the proof started long before.