Proving Convergence of \{b_n\} when \{a_n\}\to A, \{a_nb_n\} Converge

In summary, to prove the convergence of a sequence, we need to show that the terms of the sequence get closer and closer to a fixed value as the index increases. If we are given that another sequence converges to a certain value, we can use this information to show that the terms of the first sequence also converge to the same value by using the definition of convergence. The definition of convergence for a sequence is that for any small positive number ε, there exists an index N such that for all indices n greater than or equal to N, the terms of the sequence are within ε distance of the limit value L. In other words, the terms of the sequence get arbitrarily close to the limit value as the index increases. It is not necessary
  • #1
Dustinsfl
2,281
5
If [itex]\{a_n\}\to A, \ \{a_nb_n\}[/itex] converge, and [itex]A\neq 0[/itex], then prove [itex]\{b_n\}[/itex] converges.

Let [itex]\epsilon>0[/itex]. Then [itex]\exists N_1,N_2\in\mathbb{N}, \ n\geq N_1,N_2[/itex]

[tex]|a_n-A|<\frac{\epsilon}{2}[/tex]

And let [itex]\{a_nb_n\}\to AB[/itex]

So, [itex]|a_nb_n-AB|<\epsilon[/itex]

I don't know how to show b_n is < epsilon.
 
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  • #2
Hi Dustinsfl! :smile:

Hint: an(bn - B) :wink:​
 
  • #3
tiny-tim said:
Hi Dustinsfl! :smile:

Hint: an(bn - B) :wink:​

I am don't understand, so we have:

[tex](a_nb_n-a_nB)[/tex]

Ok, now what?
 
  • #4
limn->∞ :wink:
 

FAQ: Proving Convergence of \{b_n\} when \{a_n\}\to A, \{a_nb_n\} Converge

How do you prove the convergence of a sequence, given that another sequence it converges to?

To prove the convergence of a sequence, we need to show that the terms of the sequence get closer and closer to a fixed value as the index of the sequence increases. If we are given that another sequence converges to a certain value, we can use this information to show that the terms of the first sequence also converge to the same value by using the definition of convergence.

What is the definition of convergence for a sequence?

The definition of convergence for a sequence is that for any small positive number ε, there exists an index N such that for all indices n greater than or equal to N, the terms of the sequence are within ε distance of the limit value L. In other words, the terms of the sequence get arbitrarily close to the limit value as the index increases.

Is it necessary for both sequences to converge in order for their product to converge?

No, it is not necessary for both sequences to converge for their product to converge. If one sequence is convergent and the other is bounded, then their product will also converge. However, if both sequences are unbounded, then their product may not converge.

What is the relationship between the convergence of the product sequence and the convergence of the individual sequences?

If the product sequence converges, then both individual sequences must also converge. However, the converse is not always true. If both individual sequences converge, then the product sequence may or may not converge depending on the values of the sequences.

Are there any specific techniques or methods to prove the convergence of a product sequence?

Yes, there are various techniques and methods that can be used to prove the convergence of a product sequence. One method is to use the limit laws, such as the product rule, to simplify the product sequence and show that it converges. Another method is to use the Cauchy criterion, which states that a sequence is convergent if and only if it is Cauchy. We can also use the squeeze theorem to show the convergence of a product sequence by finding two other sequences that the product sequence is squeezed between, where one sequence converges to the same limit as the product sequence.

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