Suppose $(f_{n})_{n\geq 1}$ is a sequence of bounded functions on an interval $(a,b)$ (may be of infinite length) converging pointwise to a function $f$.
Can one conclude that the sequence of $L^{\infty}$ norms $\|f_{n}\|_{\infty}$ of the sequence is bounded? In other words, is it true that $\sup_{n\geq 1}\|f_{n}\|_{\infty}<\infty$?
Of course, the answer is a bieg, resounding NO! when we drop the assumption that each $f_{n}$ is bounded. Take for instance $$f_{n}(x)=\frac{1}{n(1-x)}$$ defined on $(1,2)$. Clearly $f_{n}\to 0$ pointwise on the interval $(1,2)$, however each of the terms of the sequence is unbounded.