Problem 130 - Thomas Calculus

Question

Prove that if \(\left\\{a_{n}\right\\}\) is a convergent sequence, then to every positive number \(\epsilon\) there corresponds an integer \(N\) such that for all \(m\) and \(n,\) $$ m>N \quad \text { and } \quad n>N \Rightarrow\left|a_{m}-a_{n}\right|<\epsilon $$

Step-by-Step Solution

Answer

If \( \{a_n\} \) is convergent, then beyond some integer \( N \), all terms get arbitrarily close, making the sequence Cauchy.

Step 1: Understand Convergence

A sequence \( \{a_n\} \) converges to a limit \( L \) if for every positive number \( \epsilon \), there exists a positive integer \( N \) such that for all \( n > N \), \( |a_n - L| < \epsilon/2 \). This is the standard definition of convergence.

Step 2: Apply the Convergence Definition to Two Terms

We need to show that if \( |a_n - L| < \epsilon/2 \) and \( |a_m - L| < \epsilon/2 \) for \( n, m > N \), then \( |a_m - a_n| < \epsilon \).

Step 3: Use the Triangle Inequality

Apply the triangle inequality: \( |a_m - a_n| = |a_m - L + L - a_n| \leq |a_m - L| + |a_n - L| \).

Step 4: Substitute and Simplify

Since \( |a_m - L| < \epsilon/2 \) and \( |a_n - L| < \epsilon/2 \), we have \( |a_m - a_n| \leq |a_m - L| + |a_n - L| < \epsilon/2 + \epsilon/2 = \epsilon \).

Step 5: Conclude the Proof

We have shown that for all \( m, n > N \), \( |a_m - a_n| < \epsilon \). This completes the proof, showing that the sequence is Cauchy, which implies convergence of real sequences.

Key Concepts

Cauchy sequence

A Cauchy sequence is a sequence of numbers such that the elements of the sequence become arbitrarily close to each other as the sequence progresses. When we say a sequence \( \{a_n\} \) is Cauchy, it means that for every positive number \( \epsilon \), there exists an integer \( N \) such that for all integers \( m, n > N \), the distance between \( a_m \) and \( a_n \) is less than \( \epsilon \).To fully grasp this, think of a Cauchy sequence as a series of steps where each step gets closer and closer together. No matter how small a gap \( \epsilon \) you choose, there will be a point in the sequence beyond which all pairs of terms \( a_m \) and \( a_n \) are within that gap. Therefore, a Cauchy sequence is all about the terms eventually "catching up" to each other.

• A sequence \( \{a_n\} \) is called a Cauchy sequence if \( |a_m - a_n| < \epsilon \) for any given \( \epsilon > 0 \), and \( m, n > N \).
• Convergent sequences are always Cauchy, but a Cauchy sequence in a complete space, like real numbers, is also convergent.

triangle inequality

The triangle inequality is a fundamental concept in mathematics and it is particularly useful in the study of sequences and series. It states that for any real numbers \( a, b, \) and \( c \), the sum of the lengths of any two sides of a triangle is greater than or equal to the length of the remaining side. In algebraic terms, it can be expressed as:\[|a + b| \leq |a| + |b|\]In the context of sequences, the triangle inequality helps us demonstrate the properties of Cauchy sequences. This inequality essentially provides a way to break down and compare the distances amongst terms in a sequence. It helps us rigorously establish the bounds.For example, if we have a sequence \( \{a_n\} \) that converges to a limit \( L \), applying the triangle inequality helps show:

• \(|a_m - a_n| = |a_m - L + L - a_n| \leq |a_m - L| + |a_n - L|\)
• This allows us to argue about the closeness of the sequence terms and their limits efficiently.

limit

In mathematics, a limit refers to the value that a sequence or function appears to approach as the index or input grows indefinitely larger or smaller. The concept of a limit is essential for understanding convergence and continuity in calculus and analysis.For a sequence \( \{a_n\} \) with a limit \( L \), it converges if the terms progressively get closer to \( L \) as \( n \) increases. Formally, a sequence \( \{a_n\} \) converges to \( L \) if for every number \( \epsilon > 0 \), there exists an integer \( N \) such that for all \( n > N \),\[|a_n - L| < \epsilon\]This expression means that beyond some point \( N \), the difference between the sequence term \( a_n \) and the limit \( L \) is smaller than any pre-selected positive distance \( \epsilon \).

• Limits enable us to understand the behavior and destination of sequences.
• The definition above provides a rigorous foundation for determining convergence and creating other mathematical theories.