This is a standard result which you can look up in most text books, or here on StackExchange: https://math.stackexchange.com/questions/57686/understanding...

In the end it doesn’t whether we go to infinity in one norm or the other.

Note that I am talking about finite dimensions, so I guess you didn’t mean the L^p norms or \ell^p for integrable functions or sequences but the finite-dimensional p-norms.