I want to prove that the following fake definition of limit doesn't capture the property that $f(x)$ tends to infinity as $x$ does :
$$\lim_{x\rightarrow +\infty} f(x) = +\infty \iff \forall M > 0 \exists X > 0\mid( f(x) > M \implies x > X ) $$
If I take a strictly increasing and continuos function so that $\lim_{x\rightarrow +\infty} f(x) = a $ with $a \in R$ then the definition is surely satisfied when $M < a$ . However, if I take an $M > a $ then $f(x)>M $ is always false.
Because implication is always true in this case , can I say to have proven that the definition is wrong ( because the definition would mean also the same of $\lim_{x\rightarrow +\infty} f(x) = a $ ) ?
For example, $f(x) = x^2$ doesn't satisfy your definition.
On the other hand $f(x) = 1 - e^{-x}$ satisfies your definition but $\lim_{x \to \infty} f(x) \ne +\infty$.