Prove if lim(x sub n)=x and x>0 then there exists a natural number M such that x sub n > 0 for all n> or equal to M

Any hints on how to start this or what the end result should be would be greatly appreciated, thanks.

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- Oct 10th 2008, 11:54 PMhayter221Another limit proof
Prove if lim(x sub n)=x and x>0 then there exists a natural number M such that x sub n > 0 for all n> or equal to M

Any hints on how to start this or what the end result should be would be greatly appreciated, thanks. - Oct 11th 2008, 12:01 AMCaptainBlack
if $\displaystyle {\rm{lim}}_{n \to \infty}{x_n}=x,\ \ x>0$ then for all $\displaystyle \varepsilon>0$ there exists an $\displaystyle N_{\varepsilon}$ such that:

$\displaystyle |x_n-x|<\varepsilon$ for all $\displaystyle n\ge N_{\varepsilon}$

Now choose $\displaystyle \varepsilon=x/2$ and you should find the result follows with $\displaystyle M=N_{\varepsilon}.$

CB - Oct 11th 2008, 12:08 AMhayter221
I'm not sure I follow, what do I do with the epsilon...sorry I have just begun to learn this and I haven't fully grasped the idea of the limit definition.

- Oct 11th 2008, 05:45 AMCaptainBlack
You can choose anything you like for $\displaystyle \varepsilon$, that is the definition of convergence for a sequence.

What we do is choose a particular value to ensure that $\displaystyle x_n$ is positive from $\displaystyle N_{\varepsilon}$ onwards. We do this by choosing any value of $\displaystyle \varepsilon \le x/2$ then as $\displaystyle |x_n-x|<\varepsilon$ for all $\displaystyle n \ge N_{\varepsilon}$ we have:

$\displaystyle |x_n-x|<x/2$

so:

$\displaystyle -x/2<x_n-x<x/2$

rearranging:

$\displaystyle x/2<x_n<3x/2$

for all $\displaystyle n \ge N_{\varepsilon}$

But $\displaystyle x>0$, so:

$\displaystyle x_n>0$ for all $\displaystyle n \ge N_{\varepsilon}$

CB - Oct 11th 2008, 11:38 AMhayter221
thanks, much appreciated!