Uniform convergence of $f_n(x)=\frac{x}{n}$ on $0\le x\le 1$. Proof verification

Question

I have proved that $f_{n}(x)=\frac{x}{n}$ is not uniformly convergent on $0 \leqslant x<1$. Please confirm the proof.

Let $f_{n}(x)=\frac{x}{n}$, where $0 \leqslant x<1$, then $$\sup\left\{f_{1}(x),f_2(x),\dots,f_n(x),\dots\right\}=\sup\left\{1,1/2,1/3,\dots,1/n,\dots\right\}$$

So $\displaystyle\lim_{n\longrightarrow \infty}\sup f_{n}(x)={1}$.

This shows that $f_{n}(x)$ is not uniformly convergent.

Answer 1

You should revise your proof. Note that $$\sup_{x\in[0,1)}\left|\frac{x}{n}\right|\leq \frac{1}{n}.$$

Answer 2

Your $\limsup$ isn't quite as it should be. It should rather have been $$\limsup_{n\to \infty}f_n(x)=\lim_{N\to\infty}\left(\sup_{n\geq N} f_n(x)\right)\\=\lim_{N\to \infty}\left(\sup\left\{\frac1N,\frac1{N+1},\ldots\right\}\right)$$