If $f(x) \rightarrow \infty$ as $x \rightarrow a$, and $g(x) \geq f(x)$ forall x near a. Does $g(x) \rightarrow \infty$ as $x \rightarrow a$

Question

I'm a student taking a real analysis course at university and I'm working down my problem sheet and I've been asked the question above.

I understand that if a function $f(x) \leq g(x)$ then for all $x$ near $a$ by definition $g(x)$ has to be bigger at all times therefore it must tend to infinity if $f(x)$ does. I am under the impression we should use first principles. So

$$0 < |x-a|< \delta \implies f(x) > M $$

Is it enough just to say?

$$0 < |x-a|< \delta \implies f(x) \leq g(x) > M $$

Thanks for your time! Any tips or tricks would be much appreciated!