Convergence in Probability Questions

I am stuck with this questions. I am supposed to use the following definition to prove them:

$\displaystyle X_n \stackrel{\mbox{P}}{\longrightarrow} X $ if $\displaystyle \forall \epsilon > 0$

$\displaystyle \lim_{n \to \infty} P[|X_n-X| \geq \epsilon]=0$

or equivalently

$\displaystyle \lim_{n \to \infty} P[|X_n-X| < \epsilon]=1$

1. Prove the following. Suppose $\displaystyle X_n \stackrel{\mbox{P}}{\longrightarrow} a $ and $\displaystyle g$ is a real function continuous at $\displaystyle a$. Then $\displaystyle g(X_n) \stackrel{\mbox{P}}{\longrightarrow} g(X) $

2. Let $\displaystyle \{a_n\}$ be a sequence of real numbers. Hence, we can also say that $\displaystyle \{a_n\}$ is a sequence of constant (degenerate) random variables. Let $\displaystyle a$ be a real number. Show that $\displaystyle a_n \longrightarrow a$ is equivalent to $\displaystyle a_n \stackrel{\mbox{P}}{\longrightarrow} a $

(is this proving that convergence in probability is equivalent as pointwise converge in sequences??)

Thanks in advance.