Having moved through the preceding compactness ideas, we now study total boundedness and Cauchy subsequences. The focus keyword for this lesson is total boundedness and Cauchy subsequences. This page records the definitions, theorems, proofs, examples, and practice items needed for undergraduate work in compactness and its consequences in metric spaces. :::theorem[Total Boundedness and Cauchy Subsequences] Let $Y$ be a subset of a metric space $(X,d)$. Then $Y$ is totally bounded if and only if every sequence in $Y$ contains a Cauchy subsequence. ::: :::proof Given that $Y\subseteq X$. To prove the equivalence. If $Y$ is totally bounded, repeatedly cover $Y$ by finitely many balls of radii $1,1/2,1/3,\ldots$ and choose nested subsequences lying infinitely often in one ball at each stage. The diagonal subsequence is Cauchy. Conversely, if $Y$ has no finite $\varepsilon_0$-net, one can inductively choose points mutually at least $\varepsilon_0$ apart, giving a sequence with no Cauchy subsequence. $\square$ ::: :::theorem[Compact Metric Spaces Are Totally Bounded] Let $(X,d)$ be compact. Then $X$ is totally bounded. ::: :::proof Given that $X$ is compact. To prove total boundedness. For $\varepsilon>0$, the balls $S(x,\varepsilon)$ with $x\in X$ form an open cover. A finite subcover gives a finite epsilon-net. $\square$ ::: :::corollary Every compact metric space is separable. ::: :::exercise [1] State the main definition or theorem from this lesson. [2] Give one example where the hypothesis is used. [3] Explain one common mistake related to this topic. ::: :::answer [1] See the first formal block of the lesson. [2] The examples in the lesson provide standard choices. [3] A common mistake is to apply a Euclidean compactness rule in an arbitrary metric space. ::: :::faq[Frequently Asked Questions] Q: Why is the metric important? A: Compactness, boundedness, convergence, and openness are all interpreted using the chosen metric. Q: What should students check first? A: Identify the space, the metric, and the exact compactness criterion being used. ::: :::call-to-action[Continue to Completeness and Compactness] subtitle: Review the definitions, proofs, and examples before moving to the next compactness idea. button: Next Lesson | /functional-analysis/metric-spaces/compactness-and-its-consequences-in-metric-spaces/completeness-and-compactness button-ghost: Previous Lesson | /functional-analysis/metric-spaces/compactness-and-its-consequences-in-metric-spaces/epsilon-nets-and-total-boundedness :::