Q:

negating a probability limit

I am trying to prove that if $\mathbb{P}(X_n = k) \rightarrow \mathbb{P}(X = k) $ then $X_n$ converges weakly to $X$ as $n \rightarrow \infty$, which means that $\mathbb{P}(X_n \leq b) \rightarrow \mathbb{P}(X \leq b)$ as $n \rightarrow \infty$, for all points $b$ of continuity on the CDF of of $X$. 
I wanted to try this using contraposition - I think this would be the best route to attempt the proof, however I am not sure how to formally show this as I'm not sure how to write the contrapositive statements. For example to negate the first statement do I negate one of the equalitiesin the probability statements so that I get something like this: 
$$\neg\left(\mathbb{P}(X_n = k) \rightarrow \mathbb{P}(X = k)\right) \equiv \mathbb{P}(X_n = k) \rightarrow \mathbb{P}(X \neq k) \text{ as } n \rightarrow \infty\;?$$

A:

The contra-positive is the following: If there exists $k$ such that $P(X_n=k)$ 
does not tend to $P(X=k)$ then there exists $b$ such that $P(X_n\leq b)$ 
does not tend to $P(X\leq b)$.
However this is not the best way to prove the result. Since your random variables are integer valued, $X_n=k$ iff $X_n \leq k$ and it is not true that  $X_n \leq k-1$. Hence $P(X_n=k)=P(X_n\leq k)-P(X_n \leq k-1) \to P(X\leq k)-P(X \leq k-1)=P(X=k)$

