# Analysis-Sequences

• March 19th 2013, 03:52 PM
leonhart
Analysis-Sequences
Suppose that (sn )converges to s. Prove that (sn )2 converges to s directly without using the product formula.
• March 19th 2013, 04:08 PM
Plato
Re: Analysis-Sequences
Quote:

Originally Posted by leonhart
Suppose that (sn )converges to s. Prove that (sn )2 converges to s directly without using the product formula.

First you have miss-stated it: $(s_n^2)\to s^2$.

Note that $|s_n^2-s^2|=|s_n-s||s_n+s|$.

Because $(s_n)$ is bounded, $|s_n+s|$ has a bound. Call its bound, $B+1$.

Because $(s_n)\to s$ you can make $|s_n-s|<\frac{\epsilon}{B+1}$.