Problem 131 - Thomas Calculus

Question

Uniqueness of limits Prove that limits of sequences are unique. That is, show that if \(L_{1}\) and \(L_{2}\) are numbers such that \(a_{n} \rightarrow L_{1}\) and \(a_{n} \rightarrow L_{2},\) then \(L_{1}=L_{2}\) .

Step-by-Step Solution

Answer

Limits of sequences are unique: if \(a_n \to L_1\) and \(a_n \to L_2\), then \(L_1 = L_2\).

Step 1: Understanding the Problem

We need to show that the limit of a sequence is unique. This means if a sequence \(a_n\) converges to two different numbers \(L_1\) and \(L_2\), then \(L_1\) must equal \(L_2\).

Step 2: Assume Two Limits Exists

Assume that \(a_n\) converges to both \(L_1\) and \(L_2\). That means for every \(\epsilon > 0\), there exist integers \(N_1\) and \(N_2\) such that for all \(n > N_1\), \(|a_n - L_1| < \epsilon\), and for all \(n > N_2\), \(|a_n - L_2| < \epsilon\).

Step 3: Using \(\epsilon/2\) Argument

Consider \(\epsilon = |L_1 - L_2|/2\). According to the definition of limits, there will be a point after which the terms of the sequence are within \(\epsilon/2\) of both \(L_1\) and \(L_2\).

Step 4: Deriving a Contradiction

Choose \(n > \max(N_1, N_2)\), so both inequalities \(|a_n - L_1| < \epsilon/2\) and \(|a_n - L_2| < \epsilon/2\) hold. By triangle inequality, we have \(|L_1 - L_2| = |L_1 - a_n + a_n - L_2| \leq |L_1 - a_n| + |a_n - L_2| < \epsilon/2 + \epsilon/2 = \epsilon\). This implies \(|L_1 - L_2| < |L_1 - L_2|/2\), which is a contradiction unless \(|L_1 - L_2| = 0\).

Step 5: Conclusion

Since the assumption \(L_1 eq L_2\) leads to a contradiction, the only possibility is \(L_1 = L_2\). Hence, limits of sequences are unique.

Key Concepts

Sequence Convergence

Understanding sequence convergence is essential to grasp why limits are unique. When we say a sequence converges, we mean it approaches a specific value as the sequence progresses. Imagine tracking a ball as it rolls on a path toward a resting point. The spot where it stops is like the limit of the sequence. Given a sequence \(a_n\) converging to limit \(L\), for every small value \(\epsilon > 0\), there is a point in the sequence after which all subsequent terms \(a_n\) fall within a distance \(\epsilon\) from \(L\).

Key points to remember about sequence convergence:

• Sequences have a unique limit if they converge.
• A converging sequence never drifts away from its limit as it progresses.
• The concept of \(\epsilon\) ensures that closeness to the limit can be controlled and measured precisely.

Understanding these basics helps reinforce why a sequence cannot converge to more than one limit, as it effectively 'locks in' on its destination.

Limit of a Sequence

The limit of a sequence is a fundamental mathematical concept, capturing the idea of a sequence approaching a specific point. Think of a limit as the final value a sequence aims to get as close to as possible as its terms increase. This is not just an accidental target, but a precise focus.

Key aspects of understanding the limit:

• A sequence \(a_n\) converges to a limit \(L\) if \( |a_n - L| < \epsilon\) for every \(\epsilon > 0\) beyond some term \(N\).
• The closer a term is to \(L\), the smaller the difference \(|a_n - L|\) is, emphasizing sequence stability.
• Limits reveal behavior of sequences as they proceed to infinity, often allowing simplification of complex problems.

This understanding of limits is critical for both theoretical and practical mathematical applications.

Proof by Contradiction

Proof by contradiction is a powerful tool in mathematics, especially when dealing with claims about uniqueness or existence. The central idea is to assume the opposite of what you want to prove and show it leads to an impossibility. Let's take the uniqueness of sequence limits as an example.

In the problem, if a sequence \(a_n\) supposedly converges to two different limits \(L_1\) and \(L_2\), we use proof by contradiction to address this. Starting with both limits assumed distinct, we derive logical consequences that ultimately clash, thus breaking the assumption.

Steps involved in contradiction:

• Begin by assuming the contrary of the statement you wish to prove.
• Using logical deductions, reach a contradiction with a known fact or principle (e.g., the inequality derived in our original problem).
• Since the assumption leads to a logical contradiction, it is rejected, thereby ensuring the original statement (uniqueness of limits) holds true.

This method is invaluable, serving as a robust framework for proving uniqueness and impossibility in mathematics.