Question

Show that if a set \(A\) in a metric space is bounded, so is each subset \(B \subseteq A\).

Short answer

If \( A \) is bounded, then every subset \( B \subseteq A \) is also bounded.

Step-by-step solution

Define Boundedness

First, recall that a subset \( A \) of a metric space \( (X, d) \) is said to be bounded if there exists a real number \( M > 0 \) and a point \( x_0 \in X \) such that for all \( a \in A \), \( d(x_0, a) \leq M \). This means the distance between any point in \( A \) and \( x_0 \) does not exceed \( M \).

Define the Subset B

Consider a subset \( B \) such that \( B \subseteq A \). Our goal is to show that \( B \) is also bounded under the same metric \( d \).

Apply the Definition to Subset

Since \( B \subseteq A \) and \( A \) is bounded, it follows directly from the definition that for every point \( b \in B \), \( b \in A \) as well. Therefore, the condition \( d(x_0, b) \leq M \) will hold for every \( b \in B \) because \( A \) is bounded with bound \( M \).

Conclude Boundedness for B

Thus, \( B \), being a subset of the bounded set \( A \), is also bounded because \( d(x_0, b) \leq M \) for all \( b \in B \). Therefore, we can conclude that for the same point \( x_0 \) and the same real number \( M \), \( B \) satisfies the definition of boundedness.

Key concepts

Understanding Bounded Sets

In the realm of metric spaces, a set is said to be bounded if there is a certain 'limit' that encloses it within a recognizable range. This means if you pick any point from the set, the distance to a fixed point in the space does not go beyond a specific number. Think of it as having a fenced yard. If all your toys are kept inside the yard, it's like being 'bounded.'
The formal definition says, a set \( A \) in a metric space \( (X, d) \) is bounded if there exists a real number \( M > 0 \) and a point \( x_0 \in X \) such that for every point \( a \) in \( A \), we have \( d(x_0, a) \leq M \). Here, \( M \) acts like the maximum stretch of the yard's fence, and \( x_0 \) is the central spot from where the fence stretches out.

• The concept of a 'bounded set' is crucial when discussing the limits and compactness within metric spaces.
• Bounded sets help define properties like completeness and support in proving various theorems.
• It simplifies many complex mathematical discussions by setting a clear boundary.

Examining Subsets

A subset is simply a part of a bigger set, containing some or possibly all elements of the bigger set. For example, if you have a basket of fruits, then just the apples from that basket form a subset.
When we say \( B \subseteq A \), we mean that every element in \( B \) is also in \( A \). In the context of bounded sets, if \( A \) is bounded, naturally, \( B \), being a smaller part of \( A \), inherits this property by default.

• The concept of subsets is pivotal in mathematics as it lays the groundwork for understanding more complex structures like intersections, unions, and complements.
• Understanding subsets helps in simplifying problems and breaking them down into manageable parts.
• Subsets are commonly used in proofs and arguments, particularly when demonstrating properties that smaller parts inherit from the whole.

Explaining the Distance Function

The distance function in metric spaces, often denoted as \( d(x, y) \), is a way to measure the 'far-apartness' between two points. Imagine it as a virtual ruler stretching from one point to another.
It adheres to specific properties:

• Non-negativity: The distance between any two points is zero or positive, \( d(x, y) \geq 0 \).
• Identity of indiscernibles: The distance between two points is zero if and only if they are the same point, \( d(x, y) = 0 \Rightarrow x = y \).
• Symmetry: Distance doesn't change direction, meaning \( d(x, y) = d(y, x) \).
• Triangle inequality: The distance between two points \( x \) and \( z \) will always be less than or equal to the distance from \( x \) to an intermediate point \( y \), and from \( y \) to \( z \). Mathematically, \( d(x, z) \leq d(x, y) + d(y, z) \).

Understanding the distance function is integral to explaining boundedness since it defines how far points within a metric space can spread from each other or a fixed point.