I have spent a few hours trying to find an example of a sequence of continuous functions $f_n$ $[0,1]\rightarrow \mathbb{R}$ that pointwise converge to a function $f$: $[0,1]\rightarrow \mathbb{R}$ that is unbounded.
Attempt: I have so far only managed to think of an example where $f_n:(0,1]\rightarrow \mathbb{R}$ when we define: $$f_n = \frac{n}{nx+2}$$
This sequence converges to $f=\frac{1}{x}$, which is clearly not bounded on the interval.
Edit thanks to Olivier Moschetta
My issue is when $x=0$ the sequence does not pointwise converge to a function on $[0,1]$.
Can anyone help me fix this example?