Converting Limsup/Limin Argument To Cauchy Argument For Sequence Convergence

In mathematical analysis, proving the convergence of a sequence is a fundamental task. Several techniques exist, each with its own strengths and applicability. Two powerful concepts are the limit superior (limsup) and limit inferior (liminf) of a sequence, which provide information about the eventual behavior of the sequence. Another crucial concept is that of a Cauchy sequence, where the terms of the sequence become arbitrarily close to each other as the index increases. This article explores how to convert a limsup/liminf argument into a Cauchy argument to demonstrate the convergence of a sequence, providing a deeper understanding of the relationship between these concepts.

We will delve into a specific exercise that showcases this conversion technique. Given a sequence $(x_n)_{n=1}^\infty$ that satisfies the condition $0 \leq x_{n+1} \leq x_n + \frac{1}{n^2}$ for all $n \in \mathbb{N}$, our goal is to prove that $\displaystyle\lim_{n\to\infty} x_n$ exists. This problem provides an excellent framework for illustrating how to transition from a limsup/liminf perspective to a Cauchy sequence approach, thereby establishing convergence.

The exercise we will address is as follows:

Given a sequence $(x_n)_{n=1}^\infty$ such that

$0 \leq x_{n+1} \leq x_n + \frac{1}{n^2}$ for all $n \in \mathbb{N}$,

prove that $\displaystyle\lim_{n\to\infty} x_n$ exists.

Before diving into the solution, let's make some initial observations. The given inequality suggests that the sequence $(x_n)$ is decreasing in a certain sense, but not strictly. The term $\frac{1}{n^2}$ acts as a perturbation, allowing for slight increases in the sequence terms. However, since the series $\sum_{n=1}^\infty \frac{1}{n^2}$ converges (it's a p-series with p=2 > 1), we intuitively expect the sequence to stabilize and converge to a limit. This intuition can be formalized using the concept of Cauchy sequences.

To demonstrate that the sequence $(x_n)$ converges, we will show that it is a Cauchy sequence. Recall that a sequence $(x_n)$ is Cauchy if for every $\epsilon > 0$, there exists an $N \in \mathbb{N}$ such that for all $m, n > N$, we have $|x_m - x_n| < \epsilon$. In our case, since the sequence is decreasing in a general sense, we can focus on showing that for $m > n > N$, $|x_m - x_n| = x_n - x_m < \epsilon$.

Let's consider $m > n$. We can write:

$x_{n+1} \leq x_n + \frac{1}{n^2}$

$x_{n+2} \leq x_{n+1} + \frac{1}{(n+1)^2} \leq x_n + \frac{1}{n^2} + \frac{1}{(n+1)^2}$

Continuing this pattern, we get:

$x_m \leq x_n + \frac{1}{n^2} + \frac{1}{(n+1)^2} + \cdots + \frac{1}{(m-1)^2}$

Rearranging the inequality, we have:

$x_n - x_m \geq -\sum_{k=n}^{m-1} \frac{1}{k^2}$

Since $0 \leq x_{n+1}$, each term is non-negative. Therefore, by the properties of the given inequalities:

$x_m \geq 0$

Then, for $m > n$, we can write:

$0 \leq x_n - x_m \leq \sum_{k=n}^{m-1} \frac{1}{k^2}$

Now, we need to show that the tail of the series $\sum_{k=1}^\infty \frac{1}{k^2}$ can be made arbitrarily small. This is crucial for establishing the Cauchy property.

We know that the series $\sum_{k=1}^\infty \frac{1}{k^2}$ converges. This implies that for any $\epsilon > 0$, there exists an $N \in \mathbb{N}$ such that for all $n > N$,

$\sum_{k=n}^\infty \frac{1}{k^2} < \epsilon$.

We can use the integral test to estimate the tail of the series. The function $f(x) = \frac{1}{x^2}$ is decreasing for $x \geq 1$, so we have:

$\sum_{k=n}^\infty \frac{1}{k^2} \leq \int_{n-1}^\infty \frac{1}{x^2} dx$

Evaluating the integral, we get:

$\int_{n-1}^\infty \frac{1}{x^2} dx = \left[-\frac{1}{x}\right]_{n-1}^\infty = \frac{1}{n-1}$

Thus,

$\sum_{k=n}^\infty \frac{1}{k^2} \leq \frac{1}{n-1}$

For $m > n$, we have:

$\sum_{k=n}^{m-1} \frac{1}{k^2} \leq \sum_{k=n}^\infty \frac{1}{k^2} \leq \frac{1}{n-1}$

Now, given $\epsilon > 0$, we choose $N$ such that $\frac{1}{N-1} < \epsilon$. Then, for all $m > n > N$, we have:

$0 \leq x_n - x_m \leq \sum_{k=n}^{m-1} \frac{1}{k^2} < \frac{1}{n-1} < \frac{1}{N-1} < \epsilon$

This shows that $|x_m - x_n| < \epsilon$ for all $m > n > N$, which means the sequence $(x_n)$ is a Cauchy sequence.

Since we have shown that the sequence $(x_n)$ is a Cauchy sequence in the real numbers, it follows from the completeness of the real numbers that the sequence converges. That is, there exists a real number $L$ such that:

$\displaystyle\lim_{n\to\infty} x_n = L$

This completes the proof that the sequence $(x_n)$ converges. The key to this proof was converting the initial limsup/liminf-like condition into a Cauchy argument by leveraging the convergence of the series $\sum_{n=1}^\infty \frac{1}{n^2}$.

The technique demonstrated in this problem can be generalized to other scenarios where sequences are defined by inequalities involving convergent series. The core idea is to use the convergence of the series to bound the differences between the terms of the sequence, thereby establishing the Cauchy property. This method is particularly useful when dealing with sequences that are not strictly monotonic but exhibit a form of asymptotic stability.

For instance, consider a sequence $(y_n)$ that satisfies a condition of the form:

$|y_{n+1} - y_n| \leq a_n$

where $\sum_{n=1}^\infty a_n$ converges. By applying a similar Cauchy argument, one can show that the sequence $(y_n)$ converges. This is because for $m > n$, we can write:

$|y_m - y_n| \leq |y_m - y_{m-1}| + |y_{m-1} - y_{m-2}| + \cdots + |y_{n+1} - y_n| \leq \sum_{k=n}^{m-1} a_k$

Since $\sum_{n=1}^\infty a_n$ converges, the tail of the series, $\sum_{k=n}^\infty a_k$, can be made arbitrarily small, which implies that $(y_n)$ is a Cauchy sequence and hence converges.

In this article, we have explored how to convert a limsup/liminf-style argument into a Cauchy argument to prove the convergence of a sequence. By analyzing the given inequality $0 \leq x_{n+1} \leq x_n + \frac{1}{n^2}$, we demonstrated that the sequence $(x_n)$ is a Cauchy sequence, leveraging the convergence of the series $\sum_{n=1}^\infty \frac{1}{n^2}$. This approach highlights the powerful connection between the concepts of Cauchy sequences, convergent series, and the convergence of sequences. The technique discussed can be generalized to a broader class of problems, making it a valuable tool in mathematical analysis. The ability to transition between different proof strategies, such as limsup/liminf and Cauchy arguments, enhances one's problem-solving capabilities and provides a deeper appreciation for the rich tapestry of mathematical concepts.