Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) and \(\left\{Y_{n}ight\}_{n=1}^{\infty}\) be sequences of random variables and let \(\left\{y_{n}ight\}_{n=1}^{\infty}\) be a sequence of real numbers.

a. Prove that if \(X_{n}=O_{p}\left(n^{-a}ight)\) and \(Y_{n}=o_{p}\left(n^{-b}ight)\) as \(n ightarrow \infty\), then \(X_{n} Y_{n}=\) \(o_{p}\left(n^{-(a+b)}ight)\) as \(n ightarrow \infty\).

b. Prove that if \(X_{n}=o_{p}\left(n^{-a}ight)\) and \(y_{n}=o\left(n^{-b}ight)\) as \(n ightarrow \infty\), then \(X_{n} y_{n}=\) \(o_{p}\left(n^{-(a+b)}ight)\) as \(n ightarrow \infty\).

c. Prove that if \(X_{n}=O_{p}\left(n^{-a}ight)\) and \(y_{n}=O\left(n^{-b}ight)\) as \(n ightarrow \infty\), then \(X_{n} y_{n}=\) \(O_{p}\left(n^{-(a+b)}ight)\) as \(n ightarrow \infty\).

d. Prove that if \(X_{n}=o_{p}\left(n^{-a}ight)\) and \(y_{n}=O\left(n^{-b}ight)\) as \(n ightarrow \infty\), then \(X_{n} y_{n}=\) \(o_{p}\left(n^{-(a+b)}ight)\) as \(n ightarrow \infty\).

e. Prove that if \(X_{n}=o_{p}\left(n^{-a}ight)\) and \(Y_{n}=o_{p}\left(n^{-b}ight)\) as \(n ightarrow \infty\), then \(X_{n} Y_{n}=\) \(o_{p}\left(n^{-(a+b)}ight)\) as \(n ightarrow \infty\).