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

In summary, we have proven that if a subset A is contained in a closed ball in a metric space M, then A is bounded. This is shown by considering the definition of a closed ball and a bounded set, and using the fact that the diameter of A is less than or equal to twice the radius of the closed ball in which it is contained.
  • #1
Hodgey8806
145
3

Homework Statement


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


Homework Equations


Def of closed-ball: [itex]\bar{B}[/itex]R(x) = {y[itex]\in[/itex]M:d(x,y)≤R} for some R>0
Def of bounded: A is bounded if [itex]\exists[/itex]R>0 s.t. d(x,y)≤R [itex]\forall[/itex]x,y[itex]\in[/itex]A
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 [itex]\bar{B}[/itex]R(y0) = {y[itex]\in[/itex]M:d(y,y0)≤R}, for some arbitrary fixed y0
1) If A=[itex]\phi[/itex], vacuously true.
2)Let A[itex]\subseteq[/itex][itex]\bar{B}[/itex]R(y0)
Let x1,x2[itex]\in[/itex]A.
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.
 
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  • #2
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.
 
  • #3
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!
 
  • #4
Hodgey8806 said:
Spse that A is contained in some closed ball.
Let that ball be [itex]\bar{B}[/itex]R(y0) = {y[itex]\in[/itex]M:d(y,y0)≤R}, for some arbitrary fixed y0
1) If A=[itex]\phi[/itex], vacuously true.
2)Let A[itex]\subseteq[/itex][itex]\bar{B}[/itex]R(y0)
Let x1,x2[itex]\in[/itex]A.
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.
 
  • #5
Thank you very much! I appreciate your help!
 

What does it mean for a set to be contained in a closed ball?

A set is considered to be contained in a closed ball if all of its elements are within the boundary of the ball. In other words, the distance between any element of the set and the center of the ball is less than or equal to the radius of the ball.

What is a closed ball?

A closed ball is a set of points in space that are all within a certain distance, called the radius, from a given center point. The boundary of a closed ball is defined by the surface of the ball.

Why is it important for a set to be bounded?

A bounded set is limited in size and does not extend infinitely. This is important because it allows for easier analysis and calculations. Without bounds, it would be difficult to draw any conclusions or make predictions about the set.

How can we prove that if A is contained in some closed ball, then A is bounded?

We can prove this by showing that all elements of A are within a certain distance from the center of the closed ball, which is the definition of a bounded set. This can be done using the triangle inequality or by directly calculating the distance between each element and the center of the ball.

What are some real-life examples of sets contained in closed balls?

An example of a set contained in a closed ball is a group of trees in a park that are all within a certain distance from a central fountain. Another example could be a swarm of bees flying around a beehive, where all the bees are within a certain radius from the hive. Essentially, any set of objects that can be contained within a defined boundary can be considered a set contained in a closed ball.

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