Prove that if A is contained in some closed ball, then A is bounded.

[Hodgey8806]

Homework Statement

Let M be a metric space and A\subseteqM be any subset:
Prove that if A is contained in some closed ball, then A is bounded.

Homework Equations

Def of closed-ball: \bar{B}_R(x) = {y\inM:d(x,y)≤R} for some R>0
Def of bounded: A is bounded if \existsR>0 s.t. d(x,y)≤R \forallx,y\inA
Empty set is defined to be bounded in for these problems

The Attempt at a Solution

Spse that A is contained in some closed ball.
Let that ball be \bar{B}_R(y₀) = {y\inM:d(y,y₀)≤R}, for some arbitrary fixed y₀
1) If A=\phi, vacuously true.
2)Let A\subseteq\bar{B}_R(y₀)
Let x₁,x₂\inA.
The d(x₁,x₂)≤d(x₁,y₀) + d(x₂,y₀)≤2R.
Thus, diam(A)≤2R and we see that A is bounded.
Q.E.D.

 

[HallsofIvy]

So this is homework but you don't know any relevant formulas or facts? And you haven't made any attempt to do the problem yourself? Are you sure you are taking this class!?

What is the definition of "bounded set"? What is the definition of "closed ball"? If you know those, then this should be relatively easy. If you don't, nothing I say will help you until you have looked up those definitions (different texts may have slightly different definitions- you need to know definitions you are to use.

 

[Hodgey8806]

Sorry, I accidentally pressed "Enter" after the title lol. Please, see my attempt. I'm pretty sure it is correct; but, I want to make sure that I don't have to say anymore or elaborate on the empty set bit.

Thank you!

 

[christoff]

Spse that A is contained in some closed ball.
Let that ball be \bar{B}_R(y₀) = {y\inM:d(y,y₀)≤R}, for some arbitrary fixed y₀
1) If A=\phi, vacuously true.
2)Let A\subseteq\bar{B}_R(y₀)
Let x₁,x₂\inA.
The d(x₁,x₂)≤d(x₁,y₀) + d(x₂,y₀)≤2R.
Thus, diam(A)≤2R and we see that A is bounded.
Q.E.D.

Yes, your proof is correct. I don't think you need to elaborate on the empty set case.

 

[Hodgey8806]

Thank you very much! I appreciate your help!