1. The problem statement, all variables and given/known data
If ##X=(x_n)## is a positive sequence which converges to ##x##, then ##(\sqrt {x_n})## converges to ##\sqrt x.##
2. The attempt at a solution
I was given a hint: ##\sqrt x_n -\sqrt x = \frac{x_n-x}{\sqrt x_n +\sqrt x}.##
How can I obtain that hint if it were never given to me?
My attempt:
We are given that ##\lim x_n=x.## thus given ##\epsilon>0## there exists an ##N## such that ##|x_n−x|<\epsilon## for all ##n\ge N.##
For ##\lim \sqrt {x_n}=\sqrt x##, given any ##\epsilon>0## there is an ##N## such that ##|\sqrt x_n -\sqrt x|<\epsilon##, for all ##n\ge N## [tex]|\sqrt{x_n} -\sqrt{x}|=\frac{|x_n-x|}{|\sqrt{x_n} +\sqrt{x}|} ...[/tex]
How can I finish this?
If ##X=(x_n)## is a positive sequence which converges to ##x##, then ##(\sqrt {x_n})## converges to ##\sqrt x.##
2. The attempt at a solution
I was given a hint: ##\sqrt x_n -\sqrt x = \frac{x_n-x}{\sqrt x_n +\sqrt x}.##
How can I obtain that hint if it were never given to me?
My attempt:
We are given that ##\lim x_n=x.## thus given ##\epsilon>0## there exists an ##N## such that ##|x_n−x|<\epsilon## for all ##n\ge N.##
For ##\lim \sqrt {x_n}=\sqrt x##, given any ##\epsilon>0## there is an ##N## such that ##|\sqrt x_n -\sqrt x|<\epsilon##, for all ##n\ge N## [tex]|\sqrt{x_n} -\sqrt{x}|=\frac{|x_n-x|}{|\sqrt{x_n} +\sqrt{x}|} ...[/tex]
How can I finish this?
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