# Epsilon proof trouble

1. Sep 28, 2008

### Unassuming

$$Suppose \lim_n \frac{a_n -1}{a_n +1} = 0. Prove that \lim_n a_n = 1.$$

I am trying to do the algebra so that -a_n < ?? < a_n , but I am having trouble. Am I going about this correctly?

I have also tried to solve each separate side of the inequality. I get a_n < (e+1)/(1-e), but this is not quite fitting.

Can somebody give me a clue please. Thanks

Edit: There is a hint in the book that says to set,

$$$$x_{n}=\frac{a_n-1}{a_n+1}$$$$ and then solve.

I have done this but I don't know what to do next.

Last edited: Sep 28, 2008
2. Sep 28, 2008

### HallsofIvy

Staff Emeritus
I presume the hint said set $x_n= (a_n-1)/(a_n+ 1)$ and solve for xn. When you did that what did you get? What is the limit of that as x goes to 0?

3. Sep 28, 2008

### Unassuming

I solved for a_n and I then took the lim of both sides so that

lim a_n = lim ( -x_n -1 ) / (x_n -1 ), then it was pretty straightforward.

Assuming I can take the lim of both sides. Can I do that?

4. Sep 28, 2008

### HallsofIvy

Staff Emeritus
Yes, of course. Just use the basic "rules" for limits. If the limits of an and bn exist, and the limit of bn is not 0, then the limit of an/bn exists.

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