I have a real sequence $a_{i,n}$, where $a_{i,n}=0$ for $i<n$. 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?

  • 1
    $\begingroup$ 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 $\max_n\lim_ia_{i,n}=0.$ $\endgroup$ Mar 18 at 12:50


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