Suppose that the sequence $\displaystyle \{ a_n \} $ of non-negative numbers converges to a point $\displaystyle a$. Show that $\displaystyle \{ \sqrt {a_n} \} \rightarrow \sqrt {a} $

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- Oct 3rd 2008, 04:10 PMtttcomraderShow this sequence converges
Suppose that the sequence $\displaystyle \{ a_n \} $ of non-negative numbers converges to a point $\displaystyle a$. Show that $\displaystyle \{ \sqrt {a_n} \} \rightarrow \sqrt {a} $

- Oct 3rd 2008, 04:53 PMJhevon
hint: note that $\displaystyle \sqrt{a_n} - \sqrt{a} = \frac {(\sqrt{a_n} - \sqrt{a})(\sqrt{a_n} + \sqrt{a})}{\sqrt{a_n} + \sqrt{a}} < \frac {a_n - a}{\sqrt{a}} \le \frac {|a_n - a|}{\sqrt{a}}$

of course, we want the last expression to be less than $\displaystyle \epsilon$ - Oct 4th 2008, 03:22 AMLaurent
- Oct 4th 2008, 06:37 AMOpalg