Suppose that (s_{n})converges to s. Prove that (s_{n )}^{2 }converges to s directly without using the product formula.

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- Mar 19th 2013, 02:52 PMleonhartAnalysis-Sequences
Suppose that (s

_{n})converges to s. Prove that (s_{n )}^{2 }converges to s directly without using the product formula. - Mar 19th 2013, 03:08 PMPlatoRe: Analysis-Sequences

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 $\displaystyle |s_n-s|<\frac{\epsilon}{B+1}$.