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- Homework Statement
- Using the Bolzano-Weierstrass Theorem (BWT) to prove continuity implies uniform continuity.
- Relevant Equations
- Definitions of continuity, uniform continuity and BWT á la Wikipedia.
In my textbook when proving continuity implies uniform continuity (which is very similar to the proof given here), BWT is used to find a converging subsequence. I cannot see why this is needed. Referring to the linked proof, if we open up the inequality ##|x_n-y_n|<\frac{1}{n}##, isn't by the squeeze theorem then ##(x_n-y_n)=0\iff x_n=y_n##, and so one can conclude these sequences converge to the same limit and jump the BWT step in the proof.