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Sequence limit defintion proof

  • Thread starter Lee33
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  • #1
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Homework Statement



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?
 

Answers and Replies

  • #2
arildno
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Homework Statement



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?
Remember first that your two limit definitions (in your two last lines), one for each sequence does NOT require that the "epsilon" and "N" included are the same quantities in each line. Agreed?

Secondly, in your last line, you have a fraction. Please say what we can estimate about its magnitude, with reference to the definition you set up in your first line.
 
  • #3
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Remember first that your two limit definitions (in your two last lines), one for each sequence does NOT require that the "epsilon" and "N" included are the same quantities in each line. Agreed?

Secondly, in your last line, you have a fraction. Please say what we can estimate about its magnitude, with reference to the definition you set up in your first line.
I did not understand what you meant. Can you elaborate further, please?
 
  • #4
vela
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Homework Statement



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?
Recognize that ##a-b = (\sqrt{a})^2 - (\sqrt{b})^2##.

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?
I did not understand what you meant. Can you elaborate further, please?
It would help if you'd identify what exactly you didn't understand. Right now, it just seems like you read what arildno wrote, didn't really think much about it, and simply said "I don't get it."
 
  • #5
160
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I did not understand what he meant. Which is why I need further elaboration.
 

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