Let \(\left\{f_{n}(x)ight\}_{n=1}^{\infty}\) and \(\left\{g_{n}(x)ight\}_{n=1}^{\infty}\) be sequences of real valued functions that converge pointwise to the real functions \(f\) and \(g\), respectively.

a. Prove that \(c f_{n} \xrightarrow{p w} c f\) as \(n ightarrow \infty\) where \(c\) is any real constant.

b. Prove that \(f_{n}+c \xrightarrow{p w} f+c\) as \(n ightarrow \infty\) where \(c\) is any real constant.

c. Prove that \(f_{n}+g_{n} \xrightarrow{p w} f+g\) as \(n ightarrow \infty\).

d. Prove that \(f_{n} g_{n} \xrightarrow{p w} f g\) as \(n ightarrow \infty\).

e. Suppose that \(g_{n}(x)>0\) and \(g(x)>0\) for all \(x \in \mathbb{R}\). Prove that \(f_{n} / g_{n} \xrightarrow{p w}\) \(f / g\) as \(n ightarrow \infty\).