sequences

[Sarahisme]

Hey, i am stuck on this question, although i think i have done a bit of it already...

 

here is what i have done so far....

 

we want an [math] N_\epsilon [/math] such that if [math] n \geq N [/math] then [math] |\frac{1}{x_n}-\frac{1}{x}| \leq \epsilon [/math]

 

we know (by the denition of a limit of a sequence)

[math]

|x_n - x| < \epsilon_2

[/math]

 

now

[math]

|\frac{1}{x_n}-\frac{1}{x}| = |\frac{x-x_n}{x \times x_n}| = |\frac{x_n-x}{x \times x_n}| = |x_n-x| \times |\frac{1}{x \times x_n}|

[/math]

 

so

[math]

|x_n-x| \times |\frac{1}{x \times x_n}| < \epsilon_2 \times \frac{1}{x \times x_n}

[/math]

 

 

but i am unsure of what to do next.

 

-Sarah

[matt grime]

Now you need to get a bound for |x+x_n|, what can yuo do with that? why do you have a cross in there?

[Sarahisme]

oh the cross stands for times ( i didnt know how to use the 'dot' multiplication notation)

[Sarahisme]

umm, i don't quite understand, why do i need to bound |x+x_n|?

[Sarahisme]

i got i got it! yay!

[matt grime]

you don't need to bound anything with + in it, my mistake; i read theough the post too quickly. so how did you bound x.x_n?

[Sarahisme]

umm i think i've got the proof, i'll be back in a few hours, so i'll put it up then