# Sequence proof

1. Sep 26, 2009

### koab1mjr

1. The problem statement, all variables and given/known data

Let (Tn) be a bounded sequence and let (Sn) be a sequence whose limit = 0. PRove that the limit of (Tn*Sn) = 0. I must complete the proof using only the definition of a limit of a sequence

2. Relevant equations
Let Sn be a sequence from N->R with a limit of s
For all epsilon > 0 there exists an n > N, such that |Sn - s| < epsion
The suprenum of a sequence is the least upper bound

3. The attempt at a solution

I am used to the numerical proofs showing limits I am not comfortable with manipulating these more abstract ones. I cannot solve for N so I am stuck but here is what I am working off of....

Since Tn is a bounded sequence, by the completeness axiom of R I know that a suprenum exisit and I call it U

Since Sn is a sequence that converges I know for epsilon > 0 there exist an n > N(For s) such that |Sn|< epsilon

I need to show that there is an N for any epsilon that implies |Sn*Tn - 0*U| < epsilon

I was debating proving that the Tn limit is the sup T then provoing the multiplaction of limits rule but that does not sound right there should be a direct method I need some hints on how to start this proof many thanks

2. Sep 26, 2009

### HallsofIvy

Staff Emeritus
You cannot use that "Tn limit is the sup T then proving the multiplaction of limits rule" because you do not know that Tn has a limit. All you know about Tn is that it is bounded. For example, Tn= (-1)n is bounded but does not converge.

Since Tn is bounded, let M be an upper bound for |Tn|. Now you know that $-MS_n\le T_nS_n\le MS_n$. Combine that with the fact that Sn converges to 0.

3. Sep 26, 2009

### koab1mjr

Thanks Halls for the response

Just so I got to close up the proof. I can say for a large N per the hypothesis Sn goes to zero so
MSn and -MSn go to zero.
It follows that SnTn goes to zero
and since epsilon > 0 MSn < epsilon.
am I done or is it more involved?

Am I right in saying for these more abstract sequence that you cannot really prove via the definition of a limit of a sequence. The goal is to show that something is less for any given epsilon?