Unbounded complete metric space9/25/2023 ![]() ![]() ![]() You can add more detail to the application of total boundedness if you feel that it’s necessary. If f ( x) B for all x in X, then the function is said to be bounded (from) below by B. ![]() citation needed If f is real-valued and f ( x) A for all x in X, then the function is said to be bounded (from) above by A. Statement: Suppose $(X, d)$ is a metric space that is complete, and totally bounded (i.e., for every $\epsilon > 0$, $\exists$ finitely many points $x_:k\in\Bbb N\rangle$ is Cauchy and therefore converges, since $X$ is complete. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 1 A function that is not bounded is said to be unbounded. I'm having trouble writing down the details of this proof formally. ![]()
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