F_{X,Y}(x,y) = Pr \left[ X \leq x , Y \leq y \right] \quad ,\forall (x,y) \in \mathbb{R^2}

\begin{matrix}
1. &  F_{X,Y}(-\infty,y)= lim_{x \rightarrow -\infty} F_{X,Y}(x,y) =0 , \forall  y \\
 & F_{X,Y}(x,-\infty)= lim_{y \rightarrow -\infty} F_{X,Y}(x,y) =0 , \forall  x \\
2. & F_{X,Y}(\infty,\infty)= lim_{x,y \rightarrow \infty} F_{X,Y}(x,y) =1 &  \\
3. &  \hskip{-4.5cm} \text{For any } h>0 , k>0, & \\
& Pr \left[ x < X < x+h , y < Y < y+k \right] &\\
&= F(x+h,y) - F(x,y+k) - F(x+h,y) +F(x,y) \\
4. & F(x,y) \text{ is continuous from the right in each argument, that is} &\\
& lim_{\epsilon \rightarrow 0^+} F(x+\epsilon,y) = lim_{\epsilon \rightarrow 0^+} F(x,y+\epsilon) =F(x,y)
\end{matrix}

Suppose, A_{n,y} = \{ -\infty < X \leq -n , -\infty < Y \leq y \} \\ ;\text{ where } n \text{ is a positive integer. Clearly, } A_{n,y} \downarrow as \quad  n \uparrow. \\
Thus, \quad lim_{n \rightarrow \infty} A_{n,y} = \phi \\
Now, \\
\begin{matrix}
F(-\infty,y) =lim_{n \rightarrow \infty} F(-n,y) &=&  lim_{n \rightarrow \infty} P(A_{n,y}) \\
&=& P(lim_{n \rightarrow \infty} A_{n,y} )  \\
& & \text{[by continuity theorem]} \\
&=& P (\phi) =0
\end{matrix} \\
\text{Similary, the other half of the property can be proved.}

Suppose, B_n=\{ -\infty < X \leq n , -\infty < Y \leq n \} . Clearly, B_n \uparrow as \\
n\uparrow.  Thus, lim B_n = \Omega .\text{Hence the proof.}

\begin{matrix} && Pr \left[ x < X \leq x+h , y < Y \leq y+k \right] \\
&=& F(x+h,y+k) - F(x+h,y) - F(x,y+k) + F(x,y) \\
&\geq &0 \end{matrix}

Suppose, C_n=\{ -\infty < X \leq x+ \frac{1}{n} , -\infty < Y \leq y \} \\ \text{,where n is a positive integer.Then, } \\
\underset{n \rightarrow \infty }{lim} F(x+1/n,y)  = F(x,y)