# uniform boundedness and pointwise convergence

Let { f n : A → ( X , d )│n ϵ N } be a sequence of bounded functions from a non empty set A to a metric space ( X , d ) and let f n converges point wise to a bounded function f : A → ( X , d ) . Then is it true that the sequence ( f n ) is uniformly bounded. If it is true then how to prove and if not then can we find a counter example. Please note that point wise convergence does not imply uniform convergence and hence does not imply uniform boundedness. Thanks in advance for any help.

I know that the converse of the above statement is true that is “ if the sequence of functions { f n : A → ( X , d )│n ϵ N } is uniformly bounded and converges point wise to a function f : A → ( X , d ) , then the function f : A → ( X , d ) is bounded”.

Take, for instance,$$\begin{array}{rccc}f_n\colon&\Bbb R&\longrightarrow&\Bbb R\\&x&\mapsto&\begin{cases}n&\text{ if }x=n\\0&\text{ otherwise.}\end{cases}\end{array}$$Then each $$f_n$$ is bounded and the sequence $$(f_n)_{n\in\Bbb N}$$ converges pointwise to the null function, which is bounded. However, the set $$\{f_n\mid n\in\Bbb N\}$$ is not unformly bounded.