This question might have already been asked, so feel free to direct me. In any case, I find the inequalities such as $\left|\max_{x \in [a, b]}f(x) - \max_{x \in [a, b]}g(x)\right| \leq \max_{x \in [a, b]}\left|f(x) - g(x)\right|$, for $f, g \in \mathcal{C}([a, b])$, be quite applicable even outside analysis. Therefore, do you know a cheat sheet or the like in which are listed all the inequalities (and preferably proofs) one can derive with the use of $\min, \max, \sup, \inf, \mathrm{abs}$ on finite number of elements of some vector space? The typical scalar multiplication and vector addition/subtraction & componentwise multiplication/division can be used on the vector elements.
Since of course such inequalities can be combined arbitrarily to get an infinite list, I am only interested in the most essential ones, like $\left|\max_{x \in [a, b]}f(x) - \max_{x \in [a, b]}g(x)\right| \leq \max_{x \in [a, b]}\left|f(x) - g(x)\right|$ for example.