# Homework Help: Prove that the lim of the sequence (S_n)=0

1. Oct 6, 2012

### bingo92

1. The problem statement, all variables and given/known data
suppose that (s_n) and (t_n) are sequences in which abs(s_n)≤t_n for all n and let lim(t_n)=0. Prove that lim (s_n)=0.

3. The attempt at a solution
I find absolute values to be really sketchy to work with I'm really in the dark if this is at all correct:

Let ε>0 be given, then -ε<(t_n)<ε for some n>N since the limit exists. Since -t_n≤ s_n ≤ t_n for all we can say that for -ε<-t_n≤ S_n≤t_n<ε for n>N above. hence lim(S_n)=0.

If you have any suggestion about how to deal with the absolutes in a general manner in these kind of proofs I'd be happy to hear it.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Oct 6, 2012

### Zondrina

Ahh now there's a very nice trick you can apply here. Recall the squeeze theorem works the same way it does for functions as it does for sequences.

So :

|sn| ≤ tn

Tells us that :

-tn ≤ sn ≤ tn