# Is there any relation in sequence of functions between convergence and boundedness?

a) A sequence of functions which is unbounded is neither point-wise nor uniformly convergent?

b) In case the sequence of functions is bounded below but unbounded above, or it is unbounded below but bounded above, the sequence of functions will be point-wise/uniformly convergent or not?

The sequence $\{f_n\}$ given by $f_n(x)=n\chi_{(0,n^{-1})}$ is below bounded but not above, and $f_n(x)\to 0$ for all $x$, so we can have pointwise convergence.