Convergence in Probability Questions

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

if

or equivalently

1. Prove the following. Suppose and is a real function continuous at . Then

2. Let be a sequence of real numbers. Hence, we can also say that is a sequence of constant (degenerate) random variables. Let be a real number. Show that is equivalent to

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

Thanks in advance.