Given two periodic sequences $x[n]$ and $y[n]$ with the same fundamental period $N$, it is intuitive that their multiplication is also a periodic sequence with period $N$. This stems from the fact that both $x[n]$ and $y[n]$ repeat in the same periodic window of time, which shows that their multiplication is periodic as well. My question is, what would be the mathematical proof behind this statement?
Excuse my lack of mathematical jargon, and thank you for the help!
$p$ is a period for $x$ if and only if $x[n+p]=x[n]$ for all $n$. Assume $p$ is a period for both $x$ and $y$, i.e. for all $n$ we have $x[n+p]=x[n]$ and $y[n+p]=y[n]$. If we let $z[n]=x[n]\cdot y[n]$ for all $n$, then this implies $$ z[n+p] = x[n+p]\cdot y[n+p]=x[n]\cdot y[n]=z[n]$$ this showing that $p$ is a period for $z$. (It may happen though that $z$ has additoinal smaller periods)