Proof of Limit of (Tn*Sn)=0 Using Definition

[koab1mjr]

Homework Statement

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

Homework 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

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

 

[HallsofIvy]

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

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

 

[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 S_n goes to zero so
MS_n and -MS_n go to zero.
It follows that S_nT_n goes to zero
and since epsilon > 0 MS_n < 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?