ordinary differential equations - On the Proof of the Uniqueness/Existence Theorem

I am trying to show that

$$|\underline{x}(t)-\underline{y}(t)|\leq \left|\underline{x}(t_0)-\underline{y}(t_0)\right|+\int_{t_0}^{t}\left|\underline{f}(\underline{x},s)-\underline{f}(\underline{y},s)\right| \ ds,$$
where $$\underline{x}(t)=\underline{x}(t_0)+\int_{t_0}^{t}\underline{f}(\underline{x},s) \ ds, \ \underline{y}(t)=\underline{y}(t_0)+\int_{t_0}^{t}\underline{f}(\underline{y},s) \ ds.$$ This is part of my proof for the uniqueness/existence theorem.

This is my working so far.

$$\begin{align} |\underline{x}(t)-\underline{y}(t)|&\leq |\underline{x}(t)|+\left|-\underline{y}(t)\right| \\ &= \left| \underline{x}(t_0)+\int_{t_0}^{t}\underline{f}(\underline{x},s) \ ds\right|+\left| -\underline{y}(t_0)-\int_{t_0}^{t}\underline{f}(\underline{y},s) \ ds\right| \\ &\leq |\underline{x}(t_0)|+\left|\int_{t_0}^{t}\underline{f}(\underline{x},s) \ ds\right|+\left|-\underline{y}(t_0)\right|+\left| -\int_{t_0}^{t}\underline{f}(\underline{y},s) \ ds\right| \\ \end{align}$$
I don't see how to combine the terms as desired.

Answer

Hint: $|(a+b)-(c+d)|=|(a-c)+(b-d)| \leq |a-c| +|b-d|$. (Your first step makes it impossible to complete the proof). [Take $a$ and $b$ to be the first and second terms of $x(t)$ and $c$ and $d$ to be the first and second terms of $y(t)$].