Suppose that $f:[a,b]\to\mathbb R$ and $|f(x)|\leqslant M$ on $[a,b]$, $P$ is a partition of $[a,b]$, and $c\in[a,b]$. Argue that $$\left|\mathcal U_{P\cup\{c\}}(f) - \mathcal U_P(f) \right|\leqslant 2M\cdot\operatorname{Mesh}(P) $$ and $$\left|\mathcal L_{P\cup\{c\}}(f) - \mathcal L_P(f) \right|\leqslant 2M\cdot\operatorname{Mesh}(P).$$
I understand that $2M\cdot\operatorname{Mesh}(P)$ is the 'largest' rectangle but I don't really understand how to go about proving this. It's not homework, I am really just practicing proofs, so I could use some help.