1. ## Another question about a limit of a sequence

Hey people.
I need to prove or to contradict this claim :
"If a sequence (a_n) converges to a finite limit and a_n != 0 for each n , then the limit of |a_(n+1)|/|a_n| as n->inf is smaller or equal to 1 , i.e
lim(|a_(n+1)|/|a_n|) <=1

I think that this claim is true . if a_n converges to a limit L, then of course a_(n+1) converges to L as well. Therefore, if L is not 0 , then lim(|a_(n+1)|/|a_n|) = |L|/|L| = 1

The problem here is what if L is to prove this claim is L is 0.
I think it should be done through the definition of a limit of a sequence.
I tried to play with that a little , but I didn't manage to prove that.

Any clues ?

2. ?? Try $a_{n} = \frac{n}{1-n}$ for n > 1

Not everything approaches from the top.

3. for a_n = n/(1-n),
a_(n+1) = (n+1)/(-n)
therefore
a_(n+1) / a_n = ((n+1)/(-n))/(n/(1-n)) = (-((1-n) (n+1))/n^2)

And the limit of |(-((1-n) (n+1))/n^2)| as n->inf is 1
Therefore this example does not contradict the claim.

I think that the claim is true.
I tried to prove that through the definition of limit of a sequence , but I didn't manage to.

4. Fair enough. Now prove it. Contradiction might be a useful approach.

5. I tried to use contradiction, but the point is that I need to prove here two things : First , I need to prove that lim( |a_(n+1) / a_n| ) exists and finite, and only then I need to prove that this limit is smaller or equal to 1..
I don't see how I can prove using contradiction that this limit exists and finite...

Any clues, anyone?