Define what is meant by a nowhere dense set in a metric space. State a version of the Baire Category theorem.

Let $f:[1, \infty) \rightarrow \mathbb{R}$ be a continuous function such that $f(n x) \rightarrow 0$ as $n \rightarrow \infty$ for every fixed $x \geqslant 1$. Show that $f(t) \rightarrow 0$ as $t \rightarrow \infty$.