Prove the second and third results of Theorem 1.18. That is, let (left{a_{n}ight}_{n=1}^{infty}), (left{b_{n}ight}_{n=1}^{infty},left{c_{n}ight}_{n=1}^{infty}), and (left{d_{n}ight}_{n=1}^{infty}) be

Question:

Prove the second and third results of Theorem 1.18. That is, let \(\left\{a_{n}ight\}_{n=1}^{\infty}\), \(\left\{b_{n}ight\}_{n=1}^{\infty},\left\{c_{n}ight\}_{n=1}^{\infty}\), and \(\left\{d_{n}ight\}_{n=1}^{\infty}\) be real sequences.

a. Prove that if \(a_{n}=o\left(b_{n}ight)\) and \(c_{n}=O\left(d_{n}ight)\) as \(n ightarrow \infty\) then \(a_{n} b_{n}=o\left(c_{n} d_{n}ight)\) as \(n ightarrow \infty\).

b. Prove that if \(a_{n}=O\left(b_{n}ight)\) and \(c_{n}=O\left(d_{n}ight)\) as \(n ightarrow \infty\) then \(a_{n} b_{n}=\) \(O\left(c_{n} d_{n}ight)\) as \(n ightarrow \infty\).

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