[Question]
Let a sequence (Sn) be convergent with limit of zero.  If each term 
of (Sn) is defined such that Sn = Tn+1 - Tn , is it possible to prove 
that the sequence (Tn) is convergent?  Can you prove that (Tn) is
convergent using the definition of cauchy sequences? Can you explain
why it is, or is not, possible to proves such a sequence (Tn) is
convergent?

**NOTE: By Tn+1 I mean T of sub(n+1), Not T(of sub n) + 1.

I was thinking that this might be possible to prove that (Tn) is 
convergent by using the definition of cauchy sequences, and fact that 
a sequence is cauchy iff it is convergent.  However, I am not sure if 
I allowed to set epsilon as epsilon / (m-n).

Here is my first stab:
Let a sequence (Sn) be convergent with limit of zero.  If each term 
of (Sn) is defined such that Sn = Tn+1 - Tn 

Let m>n>N
Let epislon>0

Since there exists an N such that n>N implies |Sn| < epsilon, we 
instead choose N such that n>N implies |Sn| < epsilon/(|m-n|)

|Tm - Tn| = |Tm - Tm-1 + Tm-1 - Tm-2 + Tm-2 + ... + Tn+1 - Tn |

<= |Tm - Tm-1| + |Tm-1 + Tm-2 | + ... + |Tn+1 - Tn|    
(by triangle inequality)

And so, |Tm - Tn | < |m-n|*[epsilon/(|m-n|)| = epsilon  
where n>N

It is not possible to prove that Tn is convergent: 

Tn+1 = Sn + Tn = Sn + Sn-1 + Tn-1 = ... =
     = Sn + Sn-1 + ... S1 + T1

So essentially, Tn+1 is a partial sum of the series

S1 + S2 + S3 + ... + Sn + ...

The fact that Sn -> 0 does not guarantee that the above 
series is converging, e.g. take Sn = 1/n.

Yes, we could look at it like a partial sum of a series SUM(Sn).

However, what is wrong with my logic for the construction of a cauchy sequence?
Is it because the epsilon must be fixed?

Thanks!

Yes, that's the gap in your "proof".  For any epsilon > o there exists N s.t.

n > N implies }Sn| < epsilon

but you cannot replace epsilon by an expression depending on n, such as:

  epsilon/|m-n|

That would be a case of "moving the goal posts"!

regards, mathtalk-ga