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Suppose $f: [a, b] \rightarrow \mathbb{R}$ be continuous. Prove that $f$ is uniformly continuous by characterization of sequences.

Here's how I approached the problem:

Suppose $f$ is not uniformly continuous. Then, there exists some $c > 0$ and sequences $(x_n)_{n \in \mathbb{N}}$ and $(y_n)_{n \in \mathbb{N}}$ in $A$, where $x_n - y_n \rightarrow 0$, but $|f(x_n) - f(y_n)| \ge c$.

This is where I'm a little stuck. My professor told me to look at $(x_n)$ and $(y_n)$ and determine if they have "good subsequences," but I don't quite understand what she means by that. What does it mean to look for "good subsequences"? How should I finish this proof? Any help would be greatly appreciated. Thank you.

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Choose by Bolzano a convergent subsequence $(x_{n_{k}})$ of $(x_{n})$. The corresponding subsequence $(y_{n_{k}})$ of $(y_{n})$ might not be convergent, but we can choose a further convergent subsequence $(y_{n_{k_{l}}})$ of $(y_{n_{k}})$.

Now $x_{n_{k_{l}}}\rightarrow x$ and $y_{n_{k_{l}}}\rightarrow y$ entail that $x=y$ because of that $x_{n}-y_{n}\rightarrow 0$. A contradiction will be deduced for that $|f(x_{n})-f(y_{n})|\geq c$ by the continuity of $f$.

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