The trivial zeroes of the Riemann Zeta function are located on $-2\mathbb N^*$ and they are simple. It is not difficult to see that, but the proof I have in mind is using the fact that $\xi(-2k)=\xi(1+2k)>0$ for $k\ge 1$ integer.

Well, it is not so simple, since it is using (part of) the functional equation for $\xi$, definitely not an elementary fact. Is there a simpler proof of the assertion in the title?