How do we know that if we keep going with a step of two in the natural numbers we will land on a perfect square, either odd or even?

Question

I really need to use this for another proof that I have. Been struggling with this. Can I say that it is trivial? It feels very common sense but would I need to specifically explain it if I do some proof?

Answer 1

Mh, perfect squares are natural. If you try all even [odd] naturals, you will land on all even [odd] squares. So yes, trivial.

Answer 2

You could very well call this trivial depending on the context, but you may also supply a quick proof if the situation demands it. Suppose you start at $n$ ($n\in \mathbb{N}$). Then $n^2 \geq n$. Since squaring preserves parity, you will reach $n^2$ by successively adding $2$ in a finite number of steps.

Answer 3

Suppose you are given the natural number $r$, then $r^2-r=r(r-1)$ is even and $$r^2=r+2\cdot\frac {r(r-1)}2$$ gives an explicit construction of a square which you will hit in due course.

It is trivial, as others have noted, that you will hit a square, but if you need a specific example of a square, this is an easy way ...