Question

Suppose that \(\lim _{n \rightarrow \infty} s_{n}=s\) (finite) and, for each \(\epsilon>0,\left|s_{n}-t_{n}\right|<\epsilon\) for large \(n\). Show that \(\lim _{n \rightarrow \infty} t_{n}=s\).

Short answer

Question: Prove that if the limit of the sequence \({s_n}\) as \(n\) approaches infinity is \(s\), and if for every positive number \(\epsilon\), \(|s_n - t_n| < \epsilon\) for large values of \(n\), then the limit of the sequence \({t_n}\) as \(n\) approaches infinity is also \(s\). Answer: We showed that for any \(\epsilon_1 > 0\) and for all \(n > N\), we have \(|t_n - s| < 2\epsilon_1\). Since this holds true for any \(\epsilon_1 > 0\), we can conclude that the limit of the sequence \({t_n}\) as \(n\) approaches infinity is also \(s\).

Step-by-step solution

Recall the definition of a limit of a sequence

The limit of a sequence \({x_n}\) as \(n\) approaches infinity, denoted by \(\lim_{n\rightarrow \infty}x_n = L\), means that for all \(\epsilon > 0\), there exists a natural number \(N\) such that \(|x_n - L| < \epsilon\) for all \(n > N\).

Use the given information and the limit definition

We are given that \(\lim_{n\rightarrow \infty}s_n = s\), which means that for all \(\epsilon_1 > 0\), there exists a natural number \(N_1\) such that \(|s_n - s| < \epsilon_1\) for all \(n > N_1\). We are also given that \(|s_n - t_n| < \epsilon\) for every \(\epsilon > 0\) and for large \(n\). Let's choose \(\epsilon_2=\epsilon_1\), so we have \(|s_n - t_n| < \epsilon_2\) for an appropriate large value of \(n\). Let \(N_2\) be such that \(|s_n - t_n| < \epsilon_2\) for all \(n > N_2\).

Apply the triangle inequality

Using the triangle inequality, we can write \(|t_n - s| \le |t_n - s_n| + |s_n - s|\). Both of these terms can be made smaller than \(\epsilon_2 = \epsilon_1\), as we have seen in Step 2. Thus, we can get \(|t_n - s| < \epsilon_1 + \epsilon_1 = 2\epsilon_1\). Now, choose \(N = \max\{N_1, N_2\}\). For any \(n > N\), the inequality \(|t_n - s| < 2\epsilon_1\) will hold.

Conclude that the limit of \({t_n}\) exists and equals \(s\)

Since we have shown that \(|t_n - s| < 2\epsilon_1\) for any \(\epsilon_1 > 0\) and for all \(n > N\), we can conclude by the definition of the limit of a sequence that \(\lim_{n\rightarrow \infty}t_n = s\).

Key concepts

Triangle Inequality

The Triangle Inequality is a fundamental principle in mathematics, often used in problems dealing with distances and limits of sequences. It states that for any real numbers or vectors, the absolute difference between the sum of two elements is less than or equal to the sum of their absolute differences. Mathematically, it can be expressed as follows: \[|a + b| \leq |a| + |b|.\]In the context of converging sequences, the triangle inequality helps in proving closeness between the elements of the sequence and a limiting value. For example, if we have two sequences \(s_n\) and \(t_n\) where \(|s_n - t_n| < \epsilon\) for large \(n\), it allows us to bound the sequence \(t_n\) close to \(s\). By applying the triangle inequality, you can derive a new bound:

• \(|t_n - s| \le |t_n - s_n| + |s_n - s|.\)

This step combines the differences to show that as \(s_n\) converges to \(s\), \(t_n\) also approaches \(s\). The application of this inequality is essential in the solution of sequence limit problems and establishes a connection between sequences.

Convergent Sequences

A sequence is said to be convergent if its terms approach a specific value, known as the limit, as the index \(n\) progresses to infinity. In simpler terms, as you move through the sequence, the numbers get closer and closer to a particular point. This limit concept is formalized for a sequence \(\{x_n\}\) as \(\lim_{n\rightarrow \infty} x_n = L\), where for any given small positive number \(\epsilon\), there exists a natural number \(N\) such that

• \(|x_n - L| < \epsilon\) for all \(n > N\).

In explaining convergent sequences, it's helpful to think of a graph where the plotted points begin to cluster tightly around the horizontal line representing \(L\). These convergent sequences are pivotal in understanding advanced calculus and analysis as they form the basis for further studying limits and functions. Given these properties, if you know one sequence \(s_n\) converges to \(s\) and another sequence \(t_n\) diverges at most by \(\epsilon\) from \(s_n\), you can employ this information to show that \(t_n\) must also converge to the same limit \(s\). This logical deduction underscores the connectedness of sequence limits in mathematics.

Epsilon-Delta Definition

The Epsilon-Delta (𝜖-𝛿) definition provides a rigorous framework for understanding limits in calculus, and it can be adapted for sequences as well. This concept states that for a sequence \(\{a_n\}\) to have a limit \(L\), for every real number \(\epsilon > 0\), there exists a corresponding natural number \(N\) such that:

• \(|a_n - L| < \epsilon\) whenever \(n > N\).

The job of \(\epsilon\) here is to signify any degree of smallness; we want the sequence to get within \(\epsilon\) of \(L\) for it to converge. In practical terms, think of \(𝜖\) as a shrinking band around \(L\) within which the sequence's elements must eventually lie if they are far enough along the sequence (beyond \(N\)). This approach is incredibly useful in proving the convergence of sequences, ensuring logic and structure.In the context of the problem, given that \(|s_n - t_n| < \epsilon\) for large \(n\), the epsilon-delta definition validates the argument that both sequences \(s_n\) and \(t_n\) get arbitrarily close to \(s\), thus, proving \(t_n\)'s convergence. This definition emphasizes the precision and systematic nature of mathematical analysis, making it an essential tool in demonstrating limit properties.