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## If the limit of sequence is zero then so is the limit of square roots

Solution: Let $\epsilon>0$. Since $\lim s_n=0$, there exists $N>0$ such that
$$|s_n-0|=s_n < \epsilon^2$$ for all $n>N$. Therefore $n>N$ implies that
$$|\sqrt{s_n}-0|=\sqrt{s_n}<\sqrt{\epsilon^2}=\epsilon$$ for all $n>N$ as desired. Thus $\lim s_n=0$.