Let \(\left\{x_{n}ight\}_{n=1}^{\infty}\) and \(\left\{y_{n}ight\}_{n=1}^{\infty}\) be sequences of real numbers such that

\[\lim _{n ightarrow \infty} x_{n}=x\]

and

\[\lim _{n ightarrow \infty} y_{n}=y\]

a. Prove that if \(c \in \mathbb{R}\) is a constant, then

\[\lim _{n ightarrow \infty} c x_{n}=c x\]

b. Prove that

\[\lim _{n ightarrow \infty}\left(x_{n}+y_{n}ight)=x+y\]

c. Prove that \[\lim _{n ightarrow \infty} x_{n} y_{n}=x y \]

d. Prove that \[
\lim _{n ightarrow \infty} \frac{x_{n}}{y_{n}}=\frac{x}{y}\]
where \(y_{n} eq 0\) for all \(n \in \mathbb{N}\) and \(y eq 0\).