Fix $a,b∈R$ with $a<b$. Define the sequence ${f_n}$ of functions by $f_n:[a,b]\to\mathbb{R}$ by $f_n(x)=x/n$. Prove that ${f_n}$ converges uniformly to the zero function.
I somewhat understand how to do this, but if you could explain it, that would be appreciated.
You need to show that, given $\varepsilon >0$, there exists a positive integer $N$ such that $n \ge N$ implies $|f_n(x)-0|<\varepsilon$ for every $x\in [a,b]$.
I'll give you a hint regarding how to do this. Since $a,b$ are fixed, $|f_n(x)-0|=|x/n| \le {1 \over n}\max\left \{ |a|,|b| \right \}$ for every $x \in [a,b]$.