Sequence Of Bounded Functions That Converge Pointwise To An Unbounded Function

Give an example of a sequence of bounded functions $\displaystyle f_n:[0,1]\longrightarrow\mathbb{R}$ that converge pointwise to an unbounded function $\displaystyle f$.

OK, so here's what I'm thinking. I want something that has nice behavior everywhere except for one point, where I want things to kind of go sour. I was thinking along the lines of converging to $\displaystyle f(x)=\frac{1}{x}$ but that function is not defined at 0.

Am I even thinking right on this?