Diameter of a subset of an open ball

[golriz]

Let A be a subset of a metric space such that A ⊆ B (p, r) for some p ∈ X and r > 0.
Show that diam(A) ≤ 2r.
B(p,r)=(p-r,p+r)
diam( B(p,r) )=sup{d(a,b)│a,b∈B(p,r) }=d(p-r,p+r)= 2r
Since A ⊆ B (p, r), the diameter of A is less than the diameter of B (p, r):
diam(A)≤2r

Is it true and enough? I think I have missed something in my argument.

 

[Dick]

p is a point in your space, r is a real number. You can't add them. That makes no sense at all. Try using the triangle inequality.