How do you prove triangle inequality for this metric?

real-analysis metric-spaces

Solution 1:

Hint: If $f$ is a concave function with $f(0) \ge 0$ then $f$ is subadditive. ie $$f(a+b) \le f(a) + f(b).$$

Can you prove this? Hope this helps!

Related

Proof Cassinis identity with induction and Fibonacci sequence
Isomorphism $H_n(D^n, S^{n - 1};A)$ and $A$
Pullback of locally free sheaves is locally free
Expressing an integer as the sum of three squares
Let G be an abelian group, and let a∈G. For n≥1,let G[n;a] := {x∈G:x^n =a}. Show that G[n; a] is either empty or equal to αG[n] := {αg : g ∈ G[n]}... [closed]
Proof using complex numbers
Calculating the sum of $f(x) = \sum_{n=0}^{\infty} {n \cdot 2^n \cdot x^n}$
How to prove for $s<1,|a+b|^s\le|a|^s+|b|^s$
Verify if $A * B = (A \cup B) - (A \cap B)$ forms a group, on sets from $G =\mathcal{P}(S)$.
Please recommend a good technical design tool for iPad or OS X
What is the best way to sync iCloud with other syncing services?
Why does Lion mount a network volume twice at login?