Question

Prove: If \(\lim _{n \rightarrow \infty} s_{n}=s\) and \(\lim _{n \rightarrow \infty} t_{n}=t,\) where \(s\) and \(t\) are finite, then $$ \lim _{n \rightarrow \infty}\left(s_{n}+t_{n}\right)=s+t \quad \text { and } \quad \lim _{n \rightarrow \infty}\left(s_{n}-t_{n}\right)=s-t $$

Short answer

Question: Prove that if the limit of the sequence $(s_n)$ is $s$ and the limit of the sequence $(t_n)$ is $t$, then the limit of the sum and difference of these sequences is equal to the sum and difference of their respective limits. Solution: We proved that if $\lim_{n\rightarrow\infty} s_n = s$ and $\lim_{n\rightarrow\infty} t_n = t$, then $\lim_{n\rightarrow\infty} (s_n + t_n) = s + t$ and $\lim_{n\rightarrow\infty} (s_n - t_n) = s - t$.

Step-by-step solution

Define the limit of a sequence

Before we begin, let's recall the definition of the limit of a sequence. A sequence \((a_n)\) has a limit \(a\) as \(n\) approaches infinity, denoted as \(\lim_{n\to\infty} a_n = a\), if for any \(\epsilon > 0\), there exists an \(N \in \mathbb{Z}_+\) such that \(n > N\) implies \(|a_n - a| < \epsilon\).

Prove the limit of the sum of the sequences

We want to prove that \(\lim_{n\rightarrow\infty} (s_n + t_n) = s + t\). Let \(\epsilon > 0\) be given. Since \(\lim_{n\rightarrow\infty} s_n = s\) and \(\lim_{n\rightarrow\infty} t_n = t\), there exist \(N_s, N_t \in \mathbb{Z}_+\) such that for \(n > N_s\), we have \(|s_n - s| < \frac{\epsilon}{2}\), and for \(n > N_t\), we have \(|t_n - t| < \frac{\epsilon}{2}\). Let \(N = \max(N_s, N_t)\). Then, for \(n > N\), we have $$ |\left(s_n + t_n\right) - \left(s + t\right)| = |(s_n - s) + (t_n - t)| \le |s_n - s| + |t_n - t| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon. $$ Thus, \(\lim_{n\rightarrow\infty} (s_n + t_n) = s + t\).

Prove the limit of the difference of the sequences

Next, we want to prove that \(\lim_{n\rightarrow\infty} (s_n - t_n) = s - t\). Using the same values of \(N_s\) and \(N_t\), let \(N = \max(N_s, N_t)\). Then, for \(n > N\), we have $$ |\left(s_n - t_n\right) - \left(s - t\right)| = |(s_n - s) - (t_n - t)| \le |s_n - s| + |t - t_n| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon. $$ Thus, \(\lim_{n\rightarrow\infty} (s_n - t_n) = s - t\). To summarize, we have proved that if \(\lim_{n\rightarrow\infty} s_n = s\) and \(\lim_{n\rightarrow\infty} t_n = t\), then \(\lim_{n\rightarrow\infty} (s_n + t_n) = s + t\) and \(\lim_{n\rightarrow\infty} (s_n - t_n) = s - t\).

Key concepts

Sequence Convergence

In mathematics, understanding how sequences behave as they go towards infinity is crucial. A sequence is simply a list of numbers in a specific order. We say a sequence converges to a particular number if it gets closer and closer to that number as we move further along the list. This concept is known as sequence convergence. Consider a sequence denoted as \(a_n\). If this sequence converges, it means that for any small positive number, often noted as \(\epsilon\), we can find a point beyond which all terms of the sequence are very close to the target number, which we call the limit. Simply put, as \(n\) (the position in the sequence) goes to infinity, the terms \(a_n\) get closer to the limit. This is a pivotal idea in evaluating sequences, as it helps us understand their long-term behavior.

Epsilon-Delta Definition

The epsilon-delta definition is a formal way to describe what it means for a sequence to converge to a limit. This rigorous approach involves two tiny numbers: \(\epsilon\) (epsilon) and \(N\). Here's how it goes: for every small positive number \(\epsilon\), there exists a threshold \(N\), such that for all terms in the sequence with \(n > N\), the difference between the sequence term \(a_n\) and the limit \(a\) is less than \(\epsilon\). - \(\epsilon\) represents how close we want to be to the limit.- \(N\) is the point in the sequence beyond which every term is that close.This definition ensures that no matter how small the circle of closeness we draw around the limit, eventually, the sequence terms will enter and remain within this circle. It's a robust method to mathematically prove sequence convergence and is essential for deeper analysis in calculus and mathematical analysis.

Arithmetic Operations on Limits

In calculus, we often encounter operations involving the limits of sequences. A foundational property of limits is how they behave under basic arithmetic operations. Specifically, if two sequences converge to limits \(s\) and \(t\), we have certain rules about the limits of their sums and differences. - The limit of the sum of two sequences \(s_n + t_n\) is equal to the sum of their limits \(s + t\).- Similarly, the limit of the difference \(s_n - t_n\) equals the difference of the limits \(s - t\).This behavior arises because adding or subtracting small numbers (those described by terms like \(s_n\) and \(t_n\) close to their limits) produces another small number close to the sum or difference of these limits. This concept simplifies calculating limits and is extensively used in solving problems involving sequences and series. Understanding these arithmetic operations on limits is key for working with more complex mathematical concepts and functions.