I have proved that the following function sequence is pointwise convergence with the limit function
\begin{cases} 1 & x= 0 \\ 0 & x\in (0,1] \end{cases}
where $f_{n}: [0,1] \to \mathbb R$ and $f_{n}(x)=e^{-nx}$
In order to disprove uniform convergence I chose the sequence $(\frac{1}{n})_{n}$ which converges to $0$ and did the following:
Let $\epsilon=\frac{1}{e}$ and let $N \in \mathbb N$ be arbitrary. Select $n = N$ as well as $x_{n}=\frac{1}{n}$. (Question, am I allowed to use $x$ as a sequence $x_{n}$ to disprove the sequence or does my $x$ have to be 'static', as in $x=\frac{1}{n}$)
$|f_{n}(x)-f(x)|=|e^{-n(\frac{1}{n})}-0|>=\frac{1}{e}$
Is this a correct way to disprove uniform convergence?