Hello, this is supposed to be an easy problem, but I am lost.

Let $\displaystyle X_{n}$ be a sequence of random variables with

$\displaystyle p_{n}(x) = \left\{ %always put [\] {

\begin{array}{cl}

1,& x=2+\frac{1}{n}\\

0,& elsewhere

\end{array}

\right.$

Why $\displaystyle \lim_{n\rightarrow \infty} p_{n}=0$ for all values of $\displaystyle x$? is it because x just approaches 2? how would you state that more formally?

Also, I don't get this.

The cdf of $\displaystyle X_n$ is

$\displaystyle

F_{n}(x) = \left\{

\begin{array}{cl}

0,& x < 2+\frac{1}{n}\\

1,& x \geq 2+\frac{1}{n}

\end{array}

\right.$

Why $\displaystyle \lim_{n \rightarrow \infty} F_{n}(x) = \left\{

\begin{array}{cl}

0,& x \leq 2\\

1,& x > 2

\end{array}

\right.$ ?? why the inequalities change?

Thanks in advance.