Convergent Subsequence in Closed Interval

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Theorem

Let $\left[{a . . b}\right]$ be a closed real interval.

Then every sequence of points of $\left[{a . . b}\right]$ contains a subsequence which converges to a point in $\left[{a . . b}\right]$.

Let $\left \langle {x_n} \right \rangle$ be a sequence in $\left[{a . . b}\right]$.

Since $\left[{a . . b}\right]$ is bounded, so is $\left \langle {x_n} \right \rangle$.

By the Bolzano-Weierstrass Theorem, $\left \langle {x_n} \right \rangle$ has a subsequence $\left \langle {x_{n_r}} \right \rangle$ which is convergent.

Suppose $x_{n_r} \to l$ as $n \to \infty$.

Since $a \le x_{n_r} \le b$, from Lower and Upper Bounds for Sequences it follows that $a \le l \le b$.

So $\left \langle {x_{n_r}} \right \rangle$ converges to a point in $\left[{a . . b}\right]$.

$\blacksquare$