Answer :
Final answer:
To prove that lim (a_n*b_n) = 0 for a bounded sequence (a_n) and a sequence (b_n) approaching 0, one shows that for any ε > 0, there exists an N such that for all n > N, |a_n*b_n| < ε, thus the product sequence approaches 0. The Algebraic Limit Theorem is not used because it requires convergent sequences and does not apply to sequences that are merely bounded or to limits involving infinity.
Explanation:
The original question is concerned with proving that for a bounded sequence (an) and another sequence (bn) where lim bn = 0, the product sequence (anbn) will have a limit of lim (anbn) = 0.
Since (an) is bounded, there exists a real number M such that |an| ≤ M for all n.
As (bn) approaches 0, for every ε > 0 there exists an N such that for all n > N, |bn| < ε/M.
Therefore, |anbn| ≤ M|bn| < ε, which implies that (anbn) approaches 0.
One cannot use the Algebraic Limit Theorem to prove this since it requires the sequences involved to be convergent, and the theorem may not apply if any of the limits involved are infinite or if the sequences are merely bounded but not convergent
