Prove that the lim of the sequence (S_n)=0

[bingo92]

Homework Statement

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.

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.

Homework Statement

Homework Equations

The Attempt at a Solution

 

[STEMucator]

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 :

|s_n| ≤ t_n

Tells us that :

-t_n ≤ s_n ≤ t_n