Sequence converging uniformly on a closed interval

Question

Prove that $x/n→ 0$ uniformly, as $n→ \infty$, on any closed interval $[a,b]$.

attempt: Then let $\epsilon > 0$, then there is $N \in N$ such that $n \geq N$ implies $|f_n(x) - f(x)| < \epsilon$. Let M = max{a,b} for all x in [a,b]. And choose $N > \frac{|x|}{\epsilon}$ so that $|\frac{x}{n}| \leq \frac{M}{n} \leq \frac{M}{N} = \epsilon$.

I am not sure if this is fine. Can someone please help me? any suggestion would be really appreciated. Thank you.

Answer 1

Hint:

$$\left|\frac{x}{n}-0\right| \leqslant \frac{max(|b|,|a|)}{n}$$