I have a sequence of functions $f_n(x)=x^n$ defined on the interval $[0, 1]$. How can I show that this sequence converges pointwise to $$f(x)= \begin{cases} 0 & \text{if}\, x \in [0, 1) \\ 1 &\text{if}\, x=1 \end{cases} $$
Obviously, I have to show there is some $N$ such that $|f_n(x)-f(x)|<\epsilon$ for all $x \in [0, 1]$. However, the limiting function being defined in the way that it is is giving me confusion.