Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, and let:
$$
d_2 ((x_1,y_1),(x_2,y_2)) = \left[d_X(x_1,x_2)^2 + d_Y (y_1,y_2)^2 \right]^{\frac{1}{2}}
$$
for the points $(x_1,y_1)$ and $(x_2,y_2)$ in $X \times Y$. I would like to show that the triangle inequality holds, that is:
$$
d_2 ((x_1,y_1),(x_3,y_3)) \leq d_2 ((x_1,y_1),(x_2,y_2)) + d_2 ((x_2,y_2),(x_3,y_3)) \tag{1}
$$
but I'm really stuck.
What I have tried so far:
Since $d_X$ and $d_Y$ are by definition metrics, we can use the triangle inequality to write:
\begin{equation}
\begin{aligned}
d_2 ((x_1,y_1),(x_2,y_2)) & + d_2 ((x_2,y_2),(x_3,y_3)) \\&
= \left[d_X(x_1,x_2)^2 + d_Y(y_1,y_2)^2 \right]^{\frac{1}{2}} + \left[d_X(x_2,x_3)^2 + d_Y(y_2,y_3)^2 \right]^{\frac{1}{2}} \\&
\leq \left[[d_X(x_1,x_3)+d_X(x_3,x_2)]^2 + [d_Y(y_1,y_3)+d_Y(y_3,y_2)]^2 \right]^{\frac{1}{2}} \\&
+ \left[[d_X(x_2,x_1)+d_X(x_1,x_3)]^2 + [d_Y(y_2,y_1)+d_Y(y_1,y_3)]^2 \right]^{\frac{1}{2}}
\end{aligned}
\end{equation}
where $(x_3,y_3) \in X \times Y$, and we know that:
\begin{equation}
\begin{aligned}
d_2 ((x_1,y_1),(x_3,y_3)) = \left[ d_X(x_1,x_3)^2 + d_Y(y_1,y_3)^2 \right]^{\frac{1}{2}}
\end{aligned}
\end{equation}
Then, substituting the above two equations into equation $(1)$ yields:
\begin{equation}
\begin{aligned}
\Bigl[ d_X(x_1,x_3)^2 & + d_Y(y_1,y_3)^2 \Bigr]^{\frac{1}{2}} \\&
\leq \left[[d_X(x_1,x_3)+d_X(x_3,x_2)]^2 + [d_Y(y_1,y_3)+d_Y(y_3,y_2)]^2 \right]^{\frac{1}{2}} \\&
+ \left[[d_X(x_2,x_1)+d_X(x_1,x_3)]^2 + [d_Y(y_2,y_1)+d_Y(y_1,y_3)]^2 \right]^{\frac{1}{2}}
\end{aligned}
\end{equation}
I'm not really sure how to proceed from here onwards (or perhaps I'm already on the wrong track), and any help would be much appreciated.
Best Answer
You have to prove: $$\begin{eqnarray*}d_X(x_1,x_3)^2+d_Y(y_1,y_3)^2 &\leq& d_X(x_1,x_2)^2+d_X(x_2,x_3)^2+d_Y(y_1,y_2)^2+d_Y(y_2,y_3)^2\\&+&2\sqrt{\left(d_X(x_1,x_2)^2+d_Y(y_1,y_2)^2\right)\left(d_X(x_2,x_3)^2+d_Y(y_2,y_3)^2\right)}\end{eqnarray*}$$ where the Cauchy-Schwarz inequality ensures that the last square root is greater or equal than: $$ d_X(x_1,x_2)\,d_X(x_2,x_3)+d_Y(y_1,y_2)\,d_Y(y_2,y_3)$$ hence it is sufficient to show that: $$d_X(x_1,x_3)^2+d_Y(y_1,y_3)^2\leq\left(d_X(x_1,x_2)+d_X(x_2,x_3)\right)^2+\left(d_Y(y_1,y_2)+d_Y(y_2,y_3)\right)^2$$ that just follows from the triangle inequality for $d_X$ and $d_Y$.