Consider the sequence of functions defined by:
$$f_n(x) := n\chi_{(0,\frac{1}{n}]}.$$
My question is, is this sequence of functions pointwise bounded but not uniformly bounded? For context, I have just come across these definitions in my RA class, however, no example was given to separate the two concepts.
Best Answer
Pointwise bounded means that for each point $x$, $f_n(x)$ is a bounded sequence. Indeed, for any $x$, there is some $n$, starting from which $x$ is not in the given interval, and so from that point onwards we have that $f_n(x)=0$, and so the sequence is bounded (as it only attains a finite number of values).
Uniformly bounded means that there is a uniform bound on all such sequences $f_n(x)$ - there is some $M>0$ such that $|f_n(x)|<M$ for all $x$ and for all $n$. To show that the sequence is not uniformly bounded, show that $f_n(x)$ can get arbitrarily large as $n$ grows, if you pick the right $x$. Can you find a sequence $(x_n)$ such that $f_n(x_n)$ gets arbitrarily large?