Graduate Proving Lim F(x,y) is the Distribution Function for X

[mathman]

TL;DR

Marginal distributions for multivariable

Let F(x,y) be the joint distribution for random variables X and Y (not necessarily independent). Is $lim_{y\to \infty}F(x,y)$ the distribution function for X? I believe it is. How to prove it?

 

[StoneTemplePython]

Summary: Marginal distributions for multivariable

Let F(x,y) be the joint distribution for random variables X and Y (not necessarily independent). Is $lim_{y\to \infty}F(x,y)$ the distribution function for X? I believe it is. How to prove it?

This is rather indirect, but I felt like answering this with inequalities, using a thread you were on earlier this year.
ref: https://www.physicsforums.com/threads/joint-pdf-and-its-marginals.965416/

Upper Bound
$F_{X,Y}(x,y)\leq F_X(x)$

- - - -
with $A$ the event $X \leq x$ and $B$ the event $Y \leq y$

Spoiler

$F_{X,Y}(x,y)$
$= P\big(A \cap B\big)$
$\leq P\big(A \cap B\big) + P\big(A \cap B^C\big)$
$= P\big( (A \cap B) \cup (A \cap B^C)\big )$
$= P\big( A\big )$
$=F_X(x)$

note: the upper bound does not depend on choice of yLower Bound (see link)

$F_X(x) + F_Y(y) - 1 \leq F_{X,Y}(x,y)$

putting upper and lower bounds together and taking limits, recalling that Y is a bona fide random variable, so you know its CDF tends to one

$F_X(x)$
$= \big\{ F_X(x) + 1 - 1\big\}$
$= \lim_{y \to \infty}\big\{ F_X(x) + F_Y(y) - 1\}$
$\leq \lim_{y \to \infty} F_{X,Y}(x,y)$
$\leq F_X(x)$

as desired