Given that $\sup_{i}\lim_{n}|a_{i,n}|=0$, can we find $n$ independent of $i$ so that it is less than a particular $\epsilon$?

Question

I have a sequence $a_{i,n}$. Given that $\sup_{i}\lim_{n}|a_{i,n}|=0$, if I want $|a_{i,n}|<\epsilon$ for a particular $\epsilon>0$ for all $i$, for large enough $n$, can we find a $n$ such that it is independent of $i$?

One thing I tried is to prove this by contradiction. Assume the $n$ goes to infinity when $i$ goes to infinity, but then I don't know how to proceed.

Answer 1

No, of course not: take any sequence that satisfies the given property and then redefine it by setting $a_{n, n} = n$.