$\lim_{k \rightarrow \infty} \frac{f(x)}{k}$ where $f: X \rightarrow \mathbb{R} \cup \{\infty\}$

Question

I have this line of assumptions for some exercise to be proven

"Let $X$ be a metric space and let $f: X \rightarrow \mathbb{R} \cup \{+\infty\}$ be a lower semi-continuous function which is bounded from below by $M$"

During the proof, I've countered this inequality

$$d(x, x_k) \leq \frac{1}{k} (f(x) - M + \frac{1}{k}) ~~~~~~~~~~~~~~~ \forall x \in X $$ Where, $(x_k)_k \in X$ is a sequence and $M < \infty $

My question is: Is there a way to prove that the RHS goes to 0 as $k \rightarrow \infty$ with the above assumptions? since $f(x)$ may equal $\infty$ for some $x$

Thanks in advance!