Proving a theorem about limits

In summary, to prove that the limit of (sntn) is equal to 0, you need to show that the absolute value of (sntn) is less than epsilon, given that the absolute value of (sn) is also less than epsilon. This can be done by using the triangle inequality and the fact that (tn) is a bounded sequence, with a bound of M. By showing that M times (sn) tends to 0, the limit of (sntn) can also be shown to be 0.
  • #1
mjjoga
14
0
a) Suppose that the limit as n goes to infinity sn=0. If (tn) is a bounded sequence, prove that lim(sntn)=0.
So I need to show that abs(sntn)<epsilon, and I know that abs(sn)<epsilon. I mean, I know abs(sntn)=abs(sn)abs(tn that didn't help.
I don't know how to go about this. I've tried the triangle inequality variations but that didn't seem to get me anywhere either. I feel like I'm going in circles.
can I get a hint?

thanks!
 
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  • #2
say M is a bound of the sequnce t_n, can you show M.s_n tends to zero?
 
  • #3
thanks, I finally figured it out!
mjjoga
 

FAQ: Proving a theorem about limits

What is a theorem about limits?

A theorem about limits is a mathematical statement that describes the relationship between the values of a function at certain points and its behavior at other points. It is used to prove the existence or non-existence of a limit for a function.

How do you prove a theorem about limits?

To prove a theorem about limits, you typically start by stating the theorem and its assumptions. Then, you use mathematical techniques such as the definition of a limit, algebraic manipulation, or the squeeze theorem to show that the limit exists or does not exist.

What is the importance of proving a theorem about limits?

Proving a theorem about limits is important because it allows us to understand the behavior of a function at certain points and make predictions about its behavior at other points. This is crucial in many areas of mathematics and science, such as calculus, physics, and engineering.

What are some common techniques used to prove a theorem about limits?

Some common techniques used to prove a theorem about limits include the definition of a limit, the squeeze theorem, and algebraic manipulation. Other techniques, such as the Cauchy criterion or the epsilon-delta method, may also be used depending on the specific theorem.

Can a theorem about limits be proven for all functions?

No, a theorem about limits may not apply to all functions. Each theorem has its own set of assumptions and conditions that must be met for it to be applicable. Additionally, some theorems may only apply to specific types of functions, such as continuous or differentiable functions.

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