The "standard" proof of completeness of $\ell^2-$space uses Minkowski inequality. More specifically, the proof presented in Introductory Functional Analysis with Applications with Applications by Kreyszig starts by considering a Cauchy sequence of $\ell^2$ sequences. Then it shows that, the sequences are actually Cauchy "slot-wise". From there, we could construct a new sequence. It then shows that the sequence is in $\ell^2$ as well by using Minkowski inequality.
I am wondering if there is any proof that does not use Minkowski inequality. Thank you.