I want some advice on approaching proofs of the triangular inequality for various metrics. I have a suspicion the answer is simply 'improve mathematical maturity' but hopefully there is a general procedure that this comes down to.
Examples: Proving $d(z_1,z_2)=|z_1-z_2|$ on $\Bbb C$, or $d(z_1,z_2)=\min\{|z_1|+|z_2|,|z_1-1|+|z_2-1|\}$ if $z_1\ne z_2$ and $d(z,z)=0$ on $\Bbb C$ etc. I just find triangular inequality tricky. Thanks
Metric spaces are so common place and so diverse that you can't really expect some general trick that will save you simply checking the property for any given particular definition of metric. There are some general tricks, like transforming one metric by a certain concave function always yields a new metric, but really, most often, you simply check the definition directly. Quite often it's not very hard, but sometimes, as in normed spaces where the triangle inequality is the non-trivial Minkowski inequality, you have to work harder.