I’m trying to find an example of a sequence of unbounded functions $f_n: [0,1] \rightarrow \mathbb{R}$ which converge pointwise to a bounded function $f: [0,1] \rightarrow \mathbb{R} $.
I can find lots of examples of bounded sequences which converge pointwise to an unbounded function but can’t think of a function which converges unbounded $\rightarrow$ bounded.
Is this possible?
Yes, it is possible, but you will have to consider functions which are not continuous, since a continuous function is bounded on a compact set by Weierstrass' theorem.
Let $g : [0,1] \to \mathbb{R}$ be an unbounded non-negative function.
For every $n\geq 0$, let $f_n = g \mathbb{1}_{g(x)> n}$.
Then for every $x\in [0,1]$, for every $k\geq g(x)$, we have $f_k(x) = g(x) * \mathbb{1}_{g(x)> k}(x) = 0$, so the sequence of functions $(f_n)_{n\in{\mathbb{N}}}$ converges pointwise to the zero function.