Show this sequence converges

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October 3rd 2008, 04:10 PM
tttcomrader

Show this sequence converges

Suppose that the sequence $\{ a_n \}$ of non-negative numbers converges to a point $a$ . Show that $\{ \sqrt {a_n} \} \rightarrow \sqrt {a}$
October 3rd 2008, 04:53 PM
Jhevon

Quote:

Originally Posted by tttcomrader

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

hint: note that $\sqrt{a_n} - \sqrt{a} = \frac {(\sqrt{a_n} - \sqrt{a})(\sqrt{a_n} + \sqrt{a})}{\sqrt{a_n} + \sqrt{a}} < \frac {a_n - a}{\sqrt{a}} \le \frac {|a_n - a|}{\sqrt{a}}$

of course, we want the last expression to be less than $\epsilon$
October 4th 2008, 03:22 AM
Laurent

Quote:

Originally Posted by Jhevon

hint: note that $\sqrt{a_n} - \sqrt{a} = \frac {(\sqrt{a_n} - \sqrt{a})(\sqrt{a_n} + \sqrt{a})}{\sqrt{a_n} + \sqrt{a}} < \frac {a_n - a}{\sqrt{a}} \le \frac {|a_n - a|}{\sqrt{a}}$

(just add absolute values from the beginning to make it quite rigorous)
October 4th 2008, 06:37 AM
Opalg

Quote:

Originally Posted by Jhevon

hint: note that $\sqrt{a_n} - \sqrt{a} = \frac {(\sqrt{a_n} - \sqrt{a})(\sqrt{a_n} + \sqrt{a})}{\sqrt{a_n} + \sqrt{a}} < \frac {a_n - a}{\sqrt{a}} \le \frac {|a_n - a|}{\sqrt{a}}$

Also note that you may have to consider the case a=0 separately.

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