I am reading Kreyszig's Intro to Functional Analysis and am a bit stoked with one of the problems (problem 12 in section 1.1, page 9):
Problem: Given a metric space $(X, d)$, show, using the general triangle inequality, that the inequality $$ |d(x,y) - d(z,w)| \leq d(x,z) + d(y,w) \tag{1}\label{1} $$ holds. The general triangle inequality is given as $$ d(x_1,x_n) \leq d(x_1,x_2) + d(x_2,x_3) + \cdots + d(x_{n-1},x_n)$$
My attempt at a solution: My reasoning is that I should take the left hand side in $\eqref{1}$ and change $d(x,y)$ into $d(x,z) + d(z,y)$ and $d(z,w)$ into $d(z,y) + d(y,w)$. Then, because of the minus sign, the $d(z,y)$ terms should disappear. So I did something like this: $$\begin{aligned} |d(x,y) - d(z,w)| \; &\leq d(x,y) + d(z,w) \\ &\leq d(x,z) + d(z,y) + d(z,y) + d(y,w) \\ &\leq d(x,z) + d(y,w) + 2d(z,y) \end{aligned}$$
Clearly, the $2d(z,y)$ term is a problem unless $z=y$ which isn't necessarily true. Maybe I shouldn't have gotten rid of the minus sign so early. Then $$\begin{aligned} |d(x,y) - d(z,w)| \; &\leq |d(x,z) + d(z,y) - d(z,y) - d(y,w)| \\ &\leq |d(x,z) - d(y,w)| \\ &\leq d(x,z) + d(y,w) \end{aligned}$$
This would be nice but I am having trouble justifying the first line of this second method. Since $d(a,b)$ is nonnegative by definition, it is not apparent to me why this would work. In fact, comparing the second line with the left hand side, it seems rather dubious. Ideas?