Suppose that (s_{n} )converges to s. Prove that (s_{n )}^{2 }converges to s directly without using the product formula.
First you have miss-stated it: $\displaystyle (s_n^2)\to s^2$.
Note that $\displaystyle |s_n^2-s^2|=|s_n-s||s_n+s|$.
Because $\displaystyle (s_n)$ is bounded, $\displaystyle |s_n+s|$ has a bound. Call its bound, $\displaystyle B+1$.
Because $\displaystyle (s_n)\to s$ you can make $\displaystyle |s_n-s|<\frac{\epsilon}{B+1}$.