Intuitive proof for $\forall n\in\mathbb{N}\ \ \forall k\in\mathbb{N}, k<n \ \colon \ \ k(k+1)\not\in [n^2, n(n+1)]$

Question

I'm trying to find an intuitive and quick proof for the statement in the question.

In my exercise book it's just stated that this is so without proving it. I'm trying to find a quick proof which shows that it's obvious.

Here's my cumbersome proof which shows that the statement is true.

Lets assume $k(k+1)\in [n^2, n(n+1)]$. Then

$n^2\leq k(k+1) \leq n(n+1) =n^2 + n$ $\qquad | -n^2$

$0\leq k^2 + k -n^2 \leq n$ $\qquad\qquad\quad\qquad| - k$

$0\leq \underbrace{k^2-n^2}_{<0 \ !} \leq n-k$

Since we get to the contradiciton $0<0$ we know that the upper statement must be true.

How would you prove this in a shorter and more intuitive way?

Answer 1

If $k < n$, then $k \cdot n < n^2$. It also obviously implies that $k+1\leq n$.

These two facts implies $$ k \cdot(k+1) \leq k \cdot n < n^2 $$