Does this sequence of function converges uniformly?

I was practicing for my final and stumbled on this problem.

Let $f_n:[0,1]\longrightarrow\Bbb R$ such that

$$f_n(x)=\begin{cases}nx, & 0\leq \text{ $x$ } \lt1/n\\1, & 1/n\leq\text{ $x$ }\leq 1\end{cases}$$

I could find its piecewise limit, which is

$$f(x) =\begin{cases}0 & \text{ if }x=0 \\1 & \text{ otherwise.}\end{cases}$$

I was thinking about using the $\sup|f(x)_n-f(x)|<\epsilon$

then I realized there were more than one interval of both $f_n$ and $f$.

So I used $x$ to indicate the supremum of each interval for $x=0$

Clearly $\sup|f(x)_n-f(x)|=0<\epsilon$

Since both function yield $0$

for $1/n\leq x\leq 1$

Clearly $\sup|f(x)_n-f(x)|=0<\epsilon$

Since both function yield $1$

Now I am stuck here

for $0\lt x\lt1/n$

What should I do with this interval?

And does $f(x)_n$ converges uniformly?