91Ó°ÊÓ

Sequence convergence is a fundamental concept in real analysis. It refers to the behavior of a sequence as the number of terms increases indefinitely. Specifically, a sequence \( \{a_n\} \) is said to converge if there exists a limit \( L \) such that for any given small number (often called \( \epsilon \) in mathematics), there is a point in the sequence beyond which all terms are within \( \epsilon \) of \( L \).
This means that as \( n \to \infty \), the terms \( a_n \) get arbitrarily close to \( L \).

• When the limit \( L \) exists, we say the sequence converges to \( L \).
• If no such \( L \) can be found, the sequence is divergent.

In the given exercise, the sequence \( \{x_n\} \), where \( x_{n+1} = x_n^2 \), converges to 0. As each term is the square of a smaller preceding term, all terms continue to shrink, confirming that the sequence approaches the limit of 0.

A recursive sequence is one in which each term is defined as a function of its preceding terms. The sequence \( \{x_n\} \) given by the exercise is an excellent example, as it starts with an initial term \( x_1 = \frac{1}{2} \) and each subsequent term is formulated as \( x_{n+1} = x_n^2 \).
This kind of definition allows a sequence to be built systematically:

• Start with a known value or set of values.
• Apply a defined rule or operation repeatedly to generate new terms.

Recursive sequences are not only easy to compute but also help in understanding the behavior of numerical patterns over time. Analyzing how \( \{x_n\} \) converges helps illustrate how recursive definitions can produce straightforward yet insightful results in mathematics.

A sequence is called monotonic if its terms consistently increase or decrease. In the case of the exercise, \( \{x_n\} \) is a monotonically decreasing sequence because each term is a square of the previous one, and for terms less than 1, squaring reduces their size.
Monotonic sequences have some special properties that make them easier to analyze:

• If a sequence is both bounded and monotonic, it must converge.
• Bounded means there is a limit beyond which terms cannot go, either above or below.

For \( \{x_n\} \), since each term is non-negative and the sequence is bounded below by zero, we conclude it not only decreases but also has to converge. Thus, understanding monotonicity is pivotal in establishing the behavior and eventual convergence of sequences in real analysis.