Related Reading: [[Least Upper Bound Property V.S. Completeness Defined by the Convergence of Cauchy Sequences]] ## Basics A Cauchy sequence is a concept in mathematics that helps define sequences that, intuitively, get “closer and closer together” as the sequence progresses. Named after the French mathematician Augustin-Louis Cauchy, this concept is central in understanding convergence and completeness in metric spaces, such as the real numbers, and in more abstract spaces, like Hilbert spaces. ### Definition A sequence $(x_n)$ in a metric space $M$ (a space where we can measure distances between points) is called a Cauchy sequence if, for every $\epsilon > 0$, there exists an integer $N$ such that for all $m, n > N$, the distance between $x_m$ and $x_n$ is less than $\epsilon$: $ d(x_m, x_n) < \epsilon $ where $d$ is the distance function (or metric) in $M$. ### Intuition - A Cauchy sequence is one where the terms of the sequence get “closer and closer” to each other as the sequence progresses. - Eventually, the terms of the sequence become so close that the difference between any two terms, beyond a certain point, can be made as small as desired (smaller than any positive number $\epsilon$). ### Importance of Cauchy Sequences 1. **Convergence**: - In complete metric spaces (like the real numbers, $\mathbb{R}$), every Cauchy sequence converges to a point within the space. - In spaces that are not complete (like the rational numbers, $\mathbb{Q}$), a Cauchy sequence might “approach” a limit that lies outside the space. For example, in $\mathbb{Q}$, the sequence 3.1, 3.14, 3.141, 3.1415, $\ldots$ converges to $\pi$, which is not a rational number. Hence, $\mathbb{Q}$ is not complete. 2. **Completeness**: - A space is said to be complete if every Cauchy sequence within it converges to a limit that also lies in the space. - For instance, the real numbers $\mathbb{R}$ form a complete space, meaning every Cauchy sequence in $\mathbb{R}$ has a limit within $\mathbb{R}$. 3. **Construction of Real Numbers**: - The real numbers can be constructed by completing the rational numbers with respect to Cauchy sequences. Essentially, all the “missing points” (limits of Cauchy sequences in $\mathbb{Q}$) are added to form $\mathbb{R}$. ### Example of a Cauchy Sequence Consider the sequence $(x_n)$ where $x_n = 1/n$ in the space of real numbers $\mathbb{R}$: - For any given $\epsilon > 0$, we can find an integer $N$ such that for all $m, n > N$, the difference $|x_m - x_n| < \epsilon$. - As $n$ and $m$ become large, $x_n$ and $x_m$ both get closer to 0, which is the limit of this Cauchy sequence in $\mathbb{R}$. ### Key Points to Remember - A Cauchy sequence does not require knowing the limit but rather only that terms get arbitrarily close to each other. - Completeness of a space is defined by whether every Cauchy sequence has a limit within the space. - In complete spaces, Cauchy sequences are synonymous with convergent sequences. ## Questions ### Prove that $\mathbb{R}$ is [[Completeness|complete]] (i.e. show that for all Cauchy sequences within $\mathbb{R}$, they converge) #todo