Path connected

If X and Y are path connected, show that XxY is also path connected.

For a fixed $x_0\in X$ and $y_0\in Y$
Let A be the set of all the points that join $x_0\times y_0$ by a path in $U\times V$ .

Since X and Y are both path connected, there exist $\epsilon >0$ such that $B(x,\epsilon) \ \text{and} \ B(y,\epsilon)$ , where x is in $\pi_1(A)$ and y is in $\pi_2(A)$ , contained in open sets $U\subset X$ and $V\subset Y$ , respectively.

Let $x_1\in B(x,\epsilon)$ then a path can be connected from $x_1$ to x to $x_0$ .

Similarly for y

How can now show XxY is path connected?

How can I translate this to the product topology?

Can I say:

$x_1\times y_1\in B(x\times y,\epsilon)\subset A\subset U\times V$ . Therefore the exist a path from $x_1\times y_1$ to $x\times y$ to $x_0\times y_0$ ????