Real Analysis: product of convergent sequences

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SUMMARY

The discussion centers on proving that if the product of two sequences, {an} and {bn}, converges, and {an} converges to a non-zero limit A, then {bn} must also converge. The user references the epsilon-delta definition of convergence and seeks clarification on how to approach the proof, particularly through contradiction. The key takeaway is that the convergence of the product sequence implies the convergence of the second sequence when the first sequence converges to a non-zero limit.

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Homework Statement



suppose {an} and {bn} are sequences such that {an} converges to A where A does not equal zero and {(an)(bn)} converges. prove that {bn} converges.


Homework Equations



What i have so far:
(Note:let E be epsilon)
i know that if {an} converges to A and {bn}converges to B then {(an)(bn)} converges to AB.

Let {(an)(bn)}converge to a limit, call it L. E > 0 is given, there exists a positive integer N such that n>N implies
|(an)(bn) − L| < E


The Attempt at a Solution



how can i prove that if the product of two sequences is a convergent sequence, then the two multiplies sequences are also convergent? i think i need to prove this with a contradiction but i don't know why if {an} is convergent, {bn} can't be divergent...what does it mean if it is divergent?
please help any way you can!
 
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for proof by contradiction, start with the definition of convergence and take the logical NOT of the statement
 
Last edited:

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