Folland Proves in Lemma 3.26 that if $F \in BV$, then $T\_F+F$ and $T\_F-F$ are increasing, where: [![enter image description here](https://i.stack.imgur.com/KDHkg.png)](https://i.stack.imgur.com/KDHkg.png)

I can sort of follow the proof in Folland, but I don't understand why $T\_F+F$ and $T\_F-F$ are increasing for a BV function visually or graphically. Say $sin(x)$ on $[0,2 \pi]$ is a BV function. I know that $T\_F$ is increasing, but why are $T\_F +sin(x)$ and $T\_F-sin(x)$ increasing? I can't visualize why such is true. A picture would really help me understand. Thanks. I just need more visual help for understanding the definition of total variation.