TITLE: Cartesian Product of Two Complete Metric Spaces is Complete
QUESTION [7 upvotes]: This is a problem I'm stuck on that our professor gave us for additional practice (not homework, but its recommended that we understand how to prove it). 
We know X and Y are complete metric spaces, and we need to show that $X \times Y$ is complete. I'm really lost on the proof technique. We were given an outline as follows, but I could only fully figure out (1). Part 3 is what we've been really stuck on though.  I was wondering whether someone could give an proof for say a more specific space where $X = \mathbb{R}, Y = \mathbb{R}$, so I could understand the principle.
Outlined: 
1) Show that $d_{X \times Y} ( (a_1,b_1) , (a_2,b_2)) = \max \{ d_X (a_1,a_2) , d_Y (b_1, b_2)\}$ is a metric. 
2) Prove that this gives the product topology on $X \times Y$. 
3) Prove that  if  $a_n, b_n$ are Cauchy sequences, where $a_n \in X$ and $b_n \in Y$, then $(a_n,b_n )$ is Cauchy.

REPLY [2 votes]: Actually if we want to prove that a product space is complete then we have to take a Cauchy sequence from the product space and then show that it converges to a point in it.
Observe that for all $a_1, a_2 \in X$ and $b_1, b_2 \in Y,$$d_X (a_1,a_2) \leq \max \{ d_X (a_1,a_2) , d_Y (b_1, b_2)\}\tag1$   $d_Y (b_1,b_2) \leq \max \{ d_X (a_1,a_2) , d_Y (b_1, b_2)\} \tag2$. 
Suppose we have a Cauchy sequence $((a_n, b_n))$ in $X \times Y $. Then given $\epsilon > 0$ there exist $N$ such that for all $n,m \geq N$ we have $\max \{ d_X (a_n,a_m) , d_Y (b_n, b_m)\} < \epsilon$ 
Due to $(1)$ and $(2)$ we have $d_X (a_n,a_m) < \epsilon$ and $d_Y (b_n,b_m) < \epsilon$ for all $n,m \geq N$ implying that $(a_n)$ and $(b_n)$ are Cauchy in $X$ and $Y$ respectively. Since $X$ and $Y$ are both complete $(a_n)$ converges to some $a \in X$ and $(b_n)$ converges to some $b \in Y$ and thus $(a,b) \in X \times Y.$  
Now it is a matter of showing that $((a_n, b_n))$ converges to $(a,b)$ in $X \times Y$ which is easy to do since  $\lim_{n \to \infty }d_{X \times Y} ( (a_n,b_n) , (a,b)) = \lim_{n \to \infty } \max \{ d_X (a_n,a) , d_Y (b_n, b)\} = 0.$