# continuity of maximum function with infinite many argument

I have a real sequence $$a_{i,n}$$, where $$a_{i,n}=0$$ for $$i. I want to compute $$\lim_{i\rightarrow\infty}max_{n}(a_{i,n})$$. I know that $$\lim_{i\rightarrow\infty}a_{i,n}$$ exists and bounded. I want to exchange the order of limit and maximum function. I know that maximum function is continuous if the argument is a continuous function, and also continuous for finite number of argument, but I am not sure for infinite number of argument. Under this circumstances, can I exchange the order of limit and maximum?

• Let $a_{i,i}=1+(-1)^i$ and $a_{i,j}=0$ for $i\neq j.$ Then $\max_na_{i,n}=a_{i,i}=1+(-1)^i.$ Hence the limit $\lim_i\max_na_{i,n}$ does not exist but $\max_n\lim_ia_{i,n}=0.$ Mar 18 at 12:50