I'm trying to prove proposition 4.3.7 part (d), which is: Let x,y,z,w be rational numbers. Let $\epsilon, \delta > 0$. If $|x-y| < \epsilon$ and $|z-w|<\delta$, then $|x+z|< \epsilon + \delta$, $|y+w|<\epsilon + \delta$, $|x-z|< \epsilon + \delta$, and $|y-w|< \epsilon + \delta$.
I tried to use the triangle inequality and I became stuck. I'm unsure of where to go from there. Does anyone have any suggestions?
The posted statement is wrong, as noted already in a comment.
What Proposition 4.3.7 (d) actually states is:
The above translates to $\,\big|(x+z) - (y+w)\big| \lt \epsilon+\delta\,$, which follows from the triangle inequality:
$$\,\big|(x+z) - (y+w)\big| = \big|(x-y) + (z-w)\big| \le |x-y| + |z-w| \lt \epsilon+\delta\,$$