# Is sum of two metrics a metric?

The production of two metrics is a metric also. It's googled easy. But what's about a sum? As I can see sum is metric, as the triangle inequality of metric sum is the consequence of the inequality feature and two other axioms looks obvious. Is it correct?

• yes, it is. <<characters limit for a comment>> – mm-aops Jun 6 '14 at 21:07
• Indeed, google is not a theorem prover. – Ruslan Jun 6 '14 at 21:12
• @nomen I try and described my view. – SerG Jun 6 '14 at 21:17
• You have in essence given a proof. – André Nicolas Jun 6 '14 at 21:20
• The product of metrics need not be a metric in general. The sum always is. – Henno Brandsma Sep 25 '18 at 4:49

In short, yes. Your sketched argument is correct for the triangle inequality (we're just adding two valid inequalities to get a third). One does also need to remark that the sum of two numbers $\ge 0$ can only be $0$ if both summands are $0$ (and then we apply the axiom for the two constituent metrics after that).