We need to show that a set $E \subseteq \mathbb{R}$ is bounded if and only if there exists some $r \in \mathbb{R}$ such that $x < r$ for all $x \in E$.

Step 1: Understanding boundedness

A set $E$ is said to be bounded if there exists some real number $M$ such that:

$|x| \leq M \quad \text{for all } x \in E.$

This means that every element of $E$ lies within the interval $[-M, M]$, ensuring that $E$ does not extend to infinity in either direction.

Step 2: Proving the "if" direction

Assume that there exists some $r \in \mathbb{R}$ such that:

$x < r \quad \text{for all } x \in E.$

This means that $E$ is bounded above by $r$, but we also need to show that $E$ is bounded below.

Since $E$ is a subset of $\mathbb{R}$, it must have an infimum (greatest lower bound). Let $m = \inf E$, so $m \leq x$ for all $x \in E$. Since $x < r$ for all $x \in E$, we have:

$m \leq x < r \quad \text{for all } x \in E.$

Now, define $M = \max(|m|, |r|)$. Then for all $x \in E$,

$|x| \leq M.$

Thus, $E$ is bounded.

Step 3: Proving the "only if" direction

Now, assume that $E$ is bounded, meaning there exists some $M \in \mathbb{R}$ such that:

$|x| \leq M \quad \text{for all } x \in E.$

This implies:

$-M \leq x \leq M \quad \text{for all } x \in E.$

Taking $r = M + 1$, we see that:

$x < r \quad \text{for all } x \in E.$

Thus, the existence of such an $r$ follows, completing the proof.

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Conclusion

We have shown both directions of the equivalence:

1. If there exists $r$ such that $x < r$ for all $x \in E$, then $E$ is bounded.
2. If $E$ is bounded, then there exists such an $r$.

Hence, the proof is complete.

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Would you like me to clarify any part?

Related Questions

1. How is boundedness related to the concept of supremum and infimum?
2. Can a set be bounded above but not bounded below?
3. What is the relationship between bounded sets and compactness in $\mathbb{R}$?
4. How does this result extend to sets in higher-dimensional spaces ($\mathbb{R}^n$)?
5. How does this definition of boundedness relate to metric spaces?

Tip:

A set being bounded implies the existence of a finite supremum and infimum, which is useful when working with limits and convergence in analysis.