# Bounded Subsets of a Metric Space

• gajohnson
In summary, the homework statement is saying that for a subset E of a metric space X to be bounded, there must exist a q in X and an M>0 such that d(p,q)<M for all p in E. If E is not bounded, then it cannot be demonstrated that it is compact.
gajohnson

## Homework Statement

Let X be a metric space and let E be a subset of X. Show that E is bounded if and only if there exists M>0 s.t. for all p,q in E, we have d(p,q)<M.

## Homework Equations

Use the definition of bounded which states that a subset E of a metric space X is bounded if there exists a q in X and an M>0 s.t. d(p,q)<M for all p in E.

## The Attempt at a Solution

Proof (so far):
Let E be bounded. Then there exists M/2>0 and x in X s.t. d(p,x)<M/2 for all p in E.
Now, take arbitrary p,q in E and observe that:
d(p,q) ≤ d(p,x)+d(x,q) < M/2+M/2= M
Thus, d(p,q)<M for all p,q in E.

Now, I'm getting hung up on the second part of the proof, but I feel as if it shouldn't be hard. I think I've just been staring at this for too long at this point. Any advice as to how I ought to start going in the opposite direction would be greatly appreciated. I considered trying to show that E is compact (and would therefore be closed and, more importantly, bounded), but I'm not sure that's the best route or if it's awfully easy to do.

Thanks!

Last edited:
A bounded set is not compact in general. So you won't be able to show that E is compact.

Try to pick an arbitrary element p in E. What is the distance from p to the other points in E?

micromass said:
A bounded set is not compact in general. So you won't be able to show that E is compact.

Try to pick an arbitrary element p in E. What is the distance from p to the other points in E?

You're right, not sure what I was thinking there.

Yes, as I looked at it again I just reached that conclusion before I saw your response. I suppose I overlooked that because it's so obvious!

Is it as simple as picking any q in E? Then q is in X and d(p,q)<M for all p in E.

Thanks!

Doesn't work if E is empty though, so you want to treat that in a special case.

micromass said:
Doesn't work if E is empty though, so you want to treat that in a special case.

Ah, of course.

## 1. What is a bounded subset of a metric space?

A bounded subset of a metric space is a set of points that are contained within a certain distance, known as the bound, from a fixed point in the space. The bound can be any positive real number, and it determines the size and shape of the bounded subset.

## 2. How is a bounded subset different from an unbounded subset?

A bounded subset has a finite size and is contained within a specific distance from a fixed point, while an unbounded subset has an infinite size and does not have a specific distance from a fixed point. In other words, a bounded subset is limited in size and location, while an unbounded subset has no such limitations.

## 3. What is the significance of bounded subsets in mathematics?

Bounded subsets are important in mathematics because they allow us to define and study properties of sets in a more precise way. They also have applications in various branches of mathematics, such as real analysis and topology, and are used to prove important theorems and solve problems.

## 4. Can a bounded subset have an unbounded subset?

Yes, a bounded subset can have an unbounded subset. This is because the bound only applies to the entire set, not to subsets within it. So, while the original set may be bounded, some of its subsets may not be bounded.

## 5. How do you determine if a subset is bounded in a metric space?

To determine if a subset is bounded in a metric space, you need to find a bound, or a positive real number, such that all points in the subset are within that distance from a fixed point in the space. This can be done by calculating the distance between the points in the subset and comparing them to various bound values until a suitable bound is found.

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