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More metric space help?
Exercise 5. Prove that if ρ and σ are both metrics for a set M , the
ρ + σ is also a metric for M .
I figure that like most other proofs of this variety, I am going to equate my metrics to epsilon/2, then find some way to add them together, use the triangle inequality, and then be done...
I'm just not sure which definition I'm supposed to be using in my notes.
1 Answer
- SuleimanLv 610 years agoFavorite Answer
We need to show that (M, ρ + σ) satisfies the axioms of a metric space.
For all x,y,z ∈ M we have
positivity:
(ρ + σ)(x,y) = ρ(x,y) + σ(x,y) ≥ 0 (since ρ(x,y),σ(x,y) ≥ 0);
non-degeneracy:
(ρ + σ)(x,y) = ρ(x,y) + σ(x,y) = 0 ⇒ ρ(x,y) = σ(x,y) = 0 ⇒ x = y, and
(ρ + σ)(x,x) = ρ(x,x) + σ(x,x) = 0;
symmetry:
(ρ + σ)(x,y) = ρ(x,y) + σ(x,y) = ρ(y,x) + σ(y,x) = (ρ + σ)(y,x);
triangle inequality:
(ρ + σ)(x,y) = ρ(x,y) + σ(x,y)
≤ ρ(x,z) + ρ(z,y) + σ(x,z) + σ(z,y)
= (ρ + σ)(x,z) + (ρ + σ)(z,y).
→ (M, ρ + σ) is a metric space.
