A Cauchy sequence is a sequence whose late terms become close to one another. Unlike ordinary convergence, the definition does not mention a proposed limit. This makes Cauchy sequences essential for studying completeness: a complete space is one in which every internally stabilizing sequence actually has a limit inside the space. :::definition[Cauchy Sequence] Let $(X,d)$ be a metric space. A sequence $\{x_n\}$ in $X$ is called a $Cauchy sequence$ if for every $\epsilon>0$, there exists $n_0\in\mathbb{N}$ such that $$\begin{equation*} d(x_n,x_m)<\epsilon,\qquad m,n\geq n_0. \end{equation*}$$ ::: :::example Let $x_n=1-1/2^n$. Then $\{x_n\}$ is Cauchy in $\mathbb{R}$. ::: :::proof Given that $x_n=1-1/2^n$. To prove that $\{x_n\}$ is Cauchy. For $m,n\geq n_0$, $$\begin{align*} |x_n-x_m|&=\left|\frac{1}{2^m}-\frac{1}{2^n}\right|\\ &\leq \frac{1}{2^m}+\frac{1}{2^n}\\ &\leq \frac{2}{2^{n_0}}. \end{align*}$$ Choose $n_0$ such that $2/2^{n_0}<\epsilon$. Hence $\{x_n\}$ is Cauchy. $\square$ ::: :::example The harmonic partial sums $x_n=1+1/2+\cdots+1/n$ are not Cauchy. ::: :::proof Given that $x_n=1+1/2+\cdots+1/n$. To prove that $\{x_n\}$ is not Cauchy. For $m=2n$, $$\begin{equation*} x_{2n}-x_n=\frac{1}{n+1}+\cdots+\frac{1}{2n}. \end{equation*}$$ Each term is at least $1/(2n)$, so $x_{2n}-x_n\geq 1/2$. Hence the Cauchy condition fails for $\epsilon=1/2$. $\square$ ::: :::theorem[Every Convergent Sequence Is Cauchy] Let $(X,d)$ be a metric space. If $x_n\to x$, then $\{x_n\}$ is Cauchy. ::: :::proof Given that $x_n\to x$. To prove that $\{x_n\}$ is Cauchy. Let $\epsilon>0$. Choose $n_0$ such that $d(x_n,x)<\epsilon/2$ for $n\geq n_0$. If $m,n\geq n_0$, then $$\begin{equation*} d(x_n,x_m)\leq d(x_n,x)+d(x,x_m)<\epsilon. \end{equation*}$$ Hence $\{x_n\}$ is Cauchy. $\square$ ::: :::example In $\mathbb{Q}$ with the usual metric, the decimal approximations $1.4,1.41,1.414,\ldots$ form a Cauchy sequence but do not converge in $\mathbb{Q}$, since their real limit is $\sqrt{2}$. ::: :::solved-problem Let $x_1=a$, $x_2=b$, and $x_{n+2}=(x_{n+1}+x_n)/2$. Prove that $\{x_n\}$ is Cauchy. ::: :::solution Let $x_{n+2}=(x_{n+1}+x_n)/2$. Then $$\begin{equation*} x_{n+2}-x_{n+1}=-\frac12(x_{n+1}-x_n). \end{equation*}$$ Thus $|x_{n+2}-x_{n+1}|=\frac12|x_{n+1}-x_n|$, and by induction $$\begin{equation*} |x_{n+1}-x_n|=\frac{1}{2^{n-1}}|b-a|. \end{equation*}$$ For $n>m$, $$\begin{equation*} |x_n-x_m|\leq \frac{1}{2^{m-2}}|b-a|, \end{equation*}$$ which tends to $0$ as $m\to\infty$. Hence $\{x_n\}$ is Cauchy. $\square$ ::: :::exercise [1] Define a Cauchy sequence. [2] Prove that every convergent sequence is Cauchy. [3] Give a Cauchy sequence in $\mathbb{Q}$ without rational limit. ::: :::answer [1] Late terms are mutually within every $\epsilon>0$. [2] Compare two late terms through the common limit. [3] Decimal approximations to $\sqrt{2}$. ::: :::call-to-action[Continue to Completeness] subtitle: Next, learn when every Cauchy sequence actually converges inside the same space. button: Next Lesson | /functional-analysis/metric-spaces/introduction-to-metric-spaces/complete-and-incomplete-metric-spaces button-ghost: Previous Lesson | /functional-analysis/metric-spaces/introduction-to-metric-spaces/subsequences-and-cauchy-criteria :::