Proving Convergence of Bounded Sequence {s_n*t_n} with Limit 0

In summary, if you have a sequence {s_n} of real numbers and {t_n} converges to 0, then {s_n*t_n} converges to 0.
  • #1
tarheelborn
123
0

Homework Statement


If {s_n} is a bounded sequence of real numbers and {t_n} converges to 0, prove that {s_n*t_n} converges to 0.


Homework Equations




The Attempt at a Solution


I know that since s_n is bounded, ||s_n|| <= M, but I cannot seem to find a way to make this work into the proof for convergence. Any help will be appreciated.
 
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  • #2
Since you have |sn| ≤ M what can you say about |sntn|?
 
  • #3
You can conclude that ||s_n*t_n||<=||t_n|*M. I know that t_n < epsilon, but I am having trouble working around to that.
 
  • #4
OK. Can you use that and the fact you know |tn| → 0, to claim if you were given ε > 0, you can make

|tnsn| < ε ?
 
  • #5
Thanks; you have helped immensely.
 
  • #6
Sorry, but I must still be stumped. I wrote the following:
Proof: Let \epsilon > 0. By the definition of bounded, |s_n|<=M, where M is a real number and n is an integer. This implies that |s_n*t_n|<=|t_n|*M. Since we know that t_n approaches 0 and \epsilon > 0, it follows that |s_n*t_n| < \epsilon. We can therefore conclude that s_n*t_n approaches 0. End of proof.

I can't seem to get the "we need to find a positive integer N such that |s_n*t_n|<\epsilon" part. I can't find N. I seem to be caught up in the abstractness of it. I will certainly appreciate just a tiny bit more help. Thank you.
 
  • #7
tarheelborn said:
Sorry, but I must still be stumped. I wrote the following:
Proof: Let \epsilon > 0. By the definition of bounded, |s_n|<=M, where M is a real number and n is an integer. This implies that |s_n*t_n|<=|t_n|*M. Since we know that t_n approaches 0 and \epsilon > 0, it follows that |s_n*t_n| < \epsilon. We can therefore conclude that s_n*t_n approaches 0. End of proof.

I can't seem to get the "we need to find a positive integer N such that |s_n*t_n|<\epsilon" part. I can't find N. I seem to be caught up in the abstractness of it. I will certainly appreciate just a tiny bit more help. Thank you.

I think you have most of it. First of all, if [tex]\lim t_n = 0[/tex], then what can you say about it? So, let [tex]\forall \varepsilon > 0[/tex]. Since [tex](s_n)[/tex] is bounded, then [tex]\exists M > 0[/tex] such that [tex]|s_n| \leq M[/tex] for [tex]\forall n \in \mathbb{N}[/tex]. Since [tex]\lim t_n = 0[/tex], [tex]\exists N[/tex] such that [tex]\forall n > N[/tex], we have [tex]|t_n - 0| < \varepsilon /M [/tex]. Note that we have [tex]\varepsilon / M > 0[/tex] and this is perfectly valid (and the key to why your proof is a little lacking --- can you think of why we can write [tex]\varepsilon / M[/tex] instead of just the usual [tex]\varepsilon[/tex] on the right hand side of the above?).

Then it follows that
[tex]|s_n t_n - 0| \leq M|t_n| < M \cdot \frac{\varepsilon}{M} = \varepsilon[/tex], for [tex]\forall n > N[/tex]. This completes the proof. :)
 
  • #8
Ah, yes, it does! Thank you so much. This time I really do have it! Have a nice evening and thank you.
 

What is sequence convergence?

Sequence convergence is a concept in mathematics and science that refers to the behavior of a sequence of numbers as it approaches a specific value or limit.

How is sequence convergence determined?

A sequence is said to converge if the terms in the sequence get closer and closer to a specific value as the sequence progresses. This can be determined by calculating the limit of the sequence or by observing its behavior over time.

What is the difference between sequence convergence and divergence?

Sequence divergence refers to a sequence that does not have a specific limit or value that it approaches. Instead, the terms in the sequence either get larger and larger or fluctuate with no clear pattern. In contrast, sequence convergence has a defined limit or value that the terms in the sequence approach.

What are some real-world applications of sequence convergence?

Sequence convergence is a fundamental concept in fields such as physics, engineering, and computer science. It is used to model and predict the behavior of systems and processes that involve changing quantities, such as temperature, velocity, or stock prices.

What are some common methods used to prove sequence convergence?

There are several methods used to prove sequence convergence, including the epsilon-delta method, the squeeze theorem, and the monotone convergence theorem. These methods involve analyzing the behavior of the sequence and its terms to determine if they converge to a specific limit.

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