Proof that $a^x$ goes towards infinity as x goes towards infinity

Question

I'm tasked to prove that $a^x \rightarrow \infty $ when $x \rightarrow \infty$ provided that (a > 1).

I've found a very rigorous proof for this. But my question is, why can't it be logically realized that "$a$ times $a$ an infinite amount of times produces an infinite product"?

Answer 1

If $x\in\mathbb{R}$ then

$$\forall \,N>0\,\,\exists \,M>0\,\,:\,\,\,x>M\,\,\Rightarrow \,\,a^x>N$$ because we can assume $x>\log_{a}N$. Indeed we choose $M$ such that $M>\log_{a}N$