Question

Prove: If \(s_{n} \geq 0\) for \(n \geq k\) and \(\lim _{n \rightarrow \infty} s_{n}=s,\) then \(s \geq 0\)

Short answer

Question: Prove that if a sequence \(s_n\) satisfies the condition that for any \(n \geq k\), \(s_n \geq 0\), and the limit of the sequence is \(s\), then \(s \geq 0\). Answer: One way to approach this problem is to assume the opposite, namely that \(s < 0\), and try to derive a contradiction. Using the definition of a limit, we show that for a particular choice of epsilon, there exists an \(N\) such that for all \(n > N\), we have \(s_n > \frac{s}{2}\). However, since \(s < 0\), this contradicts the given condition that \(s_n \geq 0\) for all \(n \geq k\). Therefore, our assumption must be wrong, and we conclude that \(s \geq 0\).

Step-by-step solution

Recall the definition of a limit

The definition of a limit states that for any given \(\epsilon > 0\), there exists an \(N\) such that \(n > N\) implies \(|s_n - s| < \epsilon\). This applies to our sequence and its limit \(s\).

Assume \(s < 0\) and derive a contradiction

If we assume \(s < 0\), we can consider the midpoint between \(s\) and 0, which is \(\frac{s}{2}\). Based on the definition of a limit, we can choose \(\epsilon = -\frac{s}{2} > 0\). Then there must exist an \(N\) such that for all \(n > N\), we have $$ |s_n - s| < -\frac{s}{2}. $$

Rewrite the inequality

Since \(s_n \geq 0\) for \(n \geq k\), we know that \(s_n - s \geq -s\) for \(n \geq k\). Combine this with the inequality we derived in Step 2, we have $$ -s_{n} + s< -\frac{s}{2} \implies s_n >\frac{s}{2}. $$

Show contradiction

According to the inequality derived in Step 3, we see that \(s_n > \frac{s}{2}\) for \(n > N\). However, since \(s < 0\), this implies that \(s_n > 0\), which contradicts the given condition that \(s_n \geq 0\) for \(n \geq k\).

Conclusion

Since assuming \(s < 0\) led to a contradiction, our assumption must be wrong. Therefore, the limit must be non-negative, or \(s \geq 0\). We have proven the desired result.

Key concepts

Real Analysis

Real analysis is a branch of mathematics that deals with the study of real numbers and real-valued functions. This fascinating subject focuses on concepts such as limits, continuity, derivatives, and integrals of functions of real variables. When we explore the limit of a sequence, we are diving into a fundamental concept in real analysis.

Understanding the limit of a sequence is crucial to comprehending how sequences behave as they approach infinity. This limit can be thought of as the 'destination' of the sequence—the value that the terms of the sequence get closer to, as we consider more and more terms. For example, consider the sequence \(\frac{1}{n}\) for \(n = 1, 2, 3, ...\). As \(n\) increases, the values of the sequence get closer and closer to 0, hence we say that the limit of the sequence is 0. To demonstrate this rigorously, one uses epsilon (\(\epsilon\)) arguments to show that no matter how small a positive distance you choose, eventually, all terms of the sequence fall within that distance from the supposed limit.

Convergence of Sequences

The convergence of sequences is a core concept in real analysis and pivotal for understanding many mathematical and practical problems. A sequence converges if its terms approach a specific number, called the limit, as the index goes to infinity.

To say that a sequence \(\{s_n\}\) converges to a limit \(s\), one must fulfill the formal definition: for every positive number \(\epsilon\), however small, there exists some index \(N\) such that for all \(n > N\), the terms of the sequence, \(s_n\), are within \(\epsilon\) of \(s\). This \(\epsilon-N\) definition encapsulates the intuition that terms of the sequence become arbitrarily close to \(s\), as \(n\) increases. If a sequence doesn't get closer to any single value, it is said to diverge.

Proof by Contradiction

Proof by contradiction is a powerful and often-used technique in mathematics for establishing the validity of a statement. This method begins by assuming the opposite of what you're attempting to prove. Through logical deductions, if this assumption leads to an absurdity or contradiction, one can conclude that the original assumption must be false.

In our exercise, we use proof by contradiction to demonstrate that the limit of a sequence, which is non-negative for sufficiently large indices, cannot be negative. By assuming the limit \(s\) is negative, and reaching a contradiction with the given conditions of the sequence, we are forced to reject our assumption. This mode of reasoning is universally accepted in mathematics because it hinges on the principle that a statement and its negation cannot both be true. Therefore, if the negation leads to an inconsistency, the original statement must be true. It's a logical dance that fortifies our understanding and proofs in mathematics.