The question is describes as follows:
Let $a \in \mathbb R$ and let $f$ be a function with domain $\mathbb R$. Using the formal definition of the limit, prove that: IF $\lim_{x \to a} f(x) = \infty$, THEN $\lim_{x \to a} f(x)$ diverges.
So far I can only write these two statements as formal mathematical definitions:
$\lim_{x \to a} f(x) = \infty$: $$\text{Let }a \in \mathbb R.~ \forall M \in \mathbb R,~ \exists \delta > 0 \text{ such that }0 < |x - a| < \delta \implies f(x) > M$$
$\lim_{x \to a} f(x)$ does not exist:
$$\forall L \in \mathbb R,~ \exists \epsilon > 0 \text{ such that } \forall \delta > 0,~ \exists x \in \mathbb R \text{ such that } 0 < |x-a| < \delta \implies |f(x)-L| \ge \epsilon$$
Can anyone please tell me what can I do next?