Understanding the Properties of the Interval [0,1]

The question asks about the nature of the "sequence given by interval [0,1]". While [0,1] is an interval (a set of real numbers from 0 to 1 inclusive) and not typically called a sequence, the options provided relate to properties that can apply to a set or, more commonly, a sequence whose terms are contained within that set.

The interval [0,1] consists of all real numbers $x$ such that $0 \le x \le 1$. Let's analyze the given options based on this understanding.

Analyzing the Options

1. Bounded: A set of real numbers is bounded if there is a real number $M$ (an upper bound) such that no element in the set is greater than $M$, and there is a real number $m$ (a lower bound) such that no element in the set is less than $m$. A sequence $(a_n)$ is bounded if the set $\{a_n \mid n \in \mathbb{N}\}$ is bounded.
2. Unbounded: A set or sequence is unbounded if it is not bounded, meaning it does not have either an upper bound, a lower bound, or both.
3. Convergent: A sequence $(a_n)$ is convergent if its terms approach a specific real number $L$ as $n$ goes to infinity. Convergence is a property of sequences, not typically of intervals or sets in this context.
4. Divergent: A sequence is divergent if it does not converge. Like convergence, divergence is a property of sequences.

Evaluating the Interval [0,1]

Let's consider the interval [0,1] as a set of real numbers.

• Is there an upper bound for the numbers in [0,1]? Yes, for any $x \in [0,1]$, we have $x \le 1$. So, 1 is an upper bound. Any number greater than 1 is also an upper bound.
• Is there a lower bound for the numbers in [0,1]? Yes, for any $x \in [0,1]$, we have $x \ge 0$. So, 0 is a lower bound. Any number less than 0 is also a lower bound.

Since the set of numbers in [0,1] has both an upper bound and a lower bound, the interval [0,1] is a bounded set.

If the question refers to any sequence whose terms $a_n$ are in the interval [0,1] for all $n$, i.e., $0 \le a_n \le 1$ for all $n$, then this sequence is also bounded. It is bounded below by 0 and bounded above by 1.

Conclusion on Boundedness

Whether interpreted as a set or as the range of a sequence, the interval [0,1] represents numbers that are both bounded above (by 1) and bounded below (by 0). Therefore, the property associated with the interval [0,1] among the given options is Bounded.

The concepts of convergence and divergence apply specifically to sequences and describe their behavior as the index goes to infinity. An interval like [0,1] itself does not converge or diverge.

Thus, the sequence given by the interval [0,1] (or the set of numbers in the interval) is bounded.

Revision Table: Key Concepts

Additional Information: Properties of Intervals and Sequences

Understanding the difference between sets, intervals, and sequences is crucial in real analysis. An interval like [0,1] is a continuous set of real numbers. A sequence is an ordered list of numbers, often indexed by natural numbers, e.g., $a_1, a_2, a_3, \ldots$.

A fundamental property is that every convergent sequence must be bounded. However, the converse is not true; a bounded sequence is not necessarily convergent (e.g., the sequence $a_n = (-1)^n$ is bounded between -1 and 1 but oscillates and does not converge).

The interval [0,1] itself has many properties. Besides being bounded, it is also closed (contains its boundary points 0 and 1) and connected (it's a single, unbroken segment on the real number line). These properties are central to theorems in topology and analysis, such as the Intermediate Value Theorem and the Extreme Value Theorem, which apply to continuous functions defined on closed and bounded intervals.