Let $S$ and $T$ be finite semigroups and let $(x,y)\in S\times T$. What is the index of $(x,y)$ ? Is it equals $$ \max\{\mathrm{index}(x),\mathrm{index}(y)\}\ ? $$
My proof: If $\mathrm{index}(x)=i$, $\mathrm{period}(x)=p$, $\mathrm{index}(y)=j$ and $\mathrm{period}(y)=q$, then $$ \mathrm{period}(x,y)=\mathrm{lcm}(p,q) $$ Now, if $j\leq i$, then $x^{i+\mathrm{lcm}(p,q)}=x^i$ and $y^{i+\mathrm{lcm}(p,q)}=y^i$, so $$ (x,y)^{i+\mathrm{lcm}(p,q)}=(x^{i+\mathrm{lcm}(p,q)},y^{i+\mathrm{lcm}(p,q)})=(x^i,y^i)=(x,y)^i $$ Is that proof correct?
The result is correct, but your argument needs to be a little more precise. Indeed you proved that $$ (x,y)^{i+\mathrm{lcm}(p,q)}=(x,y)^i $$ but you have to justify that $i$ is the smallest integer $s$ such that there exists $k > 0$ with $(x,y)^{s +k} =(x,y)^s$. Actually, this follows immediately from the fact that $i$ is the index of $x$, but this needs to be said to have a complete proof.