Does $f_{n}(x)=\frac{x}{n}$ converge uniformly on $[0,2]$?
Using the following definition of uniform convergence: $\forall \epsilon$,$\exists N\in \mathbb{N}$ such that if $n\geqslant N$, then $\forall x\in Domain$, $\left | f_{n}(x) - f(x) \right |<\epsilon $
Would it be correct to show that:
1) $f(x) =\lim_{n \to \infty }(\frac{x}{n})=0$
2)$\left | f_{n}(x) - f(x) \right | =\left | \frac{x}{n} - 0 \right |=\left | \frac{x}{n} \right |\leq \frac{2}{n}<\epsilon$
is true when we choose $n > \frac{\epsilon }{2}$ and thus $f_{n}(x)$ is uniformly convergent?