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If (bₙ) is a bounded sequence and lim(aₙ)=0, show that lim(aₙbₙ)=0. Explain why Theorem 3.2.3 cannot be used.
a) True
b) False

Answer :

Final answer:

True'. The statement 'If (bₙ) is a bounded sequence and lim(aₙ)=0, show that lim(aₙbₙ)=0' i strue.

The correct answer is B) true

Explanation:

If (b_n) is a bounded sequence, it means there exist real numbers M such that for all n, |b_n| ≤ M. Given that lim(a_n)=0, for every ε > 0, there exists an N such that for all n ≥ N, |a_n| < ε/M. To show that lim(a_n*b_n)=0, we use the fact that:

  • |a_n * b_n| = |a_n| * |b_n|
  • Since |b_n| ≤ M and |a_n| < ε/M for n ≥ N, then |a_n * b_n| < M * (ε/M) = ε.
  • Therefore, for all n ≥ N, |a_n * b_n| < ε, proving that lim(a_n*b_n)=0.

As for the Theorem 3.2.3, without specific information we can't precisely say why it cannot be used. It might be possible that Theorem 3.2.3 requires conditions that are not met in this scenario. For example, it might require that the sequences themselves be convergent, not just bounded, or it may require the limit of the second sequence to be non-zero for the property to hold—conditions which do not apply here.

The correct answer is B) true

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