Explanation of the Bounded Convergence Theorem

If you avoid the requirement of uniform boundedness then there is a counterexample $$ f_n=n^2 1_{[0,n^{-1}]} $$

But there are examples when the theorem holds even if the sequence of functions is not uniformly pointwise bounded. For example $$ f_n=n^{1/2}1_{[1,n^{-1}]} $$

The most general requirement on boundedness of $f_n$ when theorem still holds is $$ \forall n\in\mathbb{N}\quad\forall x\in E\quad |f_n(x)|\leq F(x) $$ for some integrable $F:E\to\mathbb{R}_+$. You can also weaken the condition of pointwise convergence just to convergence in measure $$ \forall\varepsilon>0\quad\lim\limits_{n\to\infty}\mu(\{x\in E:|f_n(x)-f(x)|>\varepsilon\})=0 $$