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**particlejohn** Prove that if $\displaystyle I_n = [a_{n}, b_{n}] $ is a nested collection of closed intervals such that $\displaystyle \lim_{n \to \infty} b_{n} - a_{n} = 0 $, then there is an $\displaystyle x \in \mathbb{R} $ such that $\displaystyle \bigcap_{n \in \mathbb{N}} I_N = \{x \} $.

So $\displaystyle a_n $ is an increasing sequence and $\displaystyle b_n $ is a decreasing sequence. So choose an $\displaystyle N \in \mathbb{N} $ so that $\displaystyle |b_{N}- a_{N}| < \varepsilon $. Then $\displaystyle a_{N} \leq a_{n} < b_{n} \leq b_{N} $ for all $\displaystyle n \geq N $. So $\displaystyle a_{n} \to \alpha $ and $\displaystyle b_n \to \beta $ for $\displaystyle \alpha, \beta \in \mathbb{R} $. We need to show that $\displaystyle x = \alpha = \beta $.

Then $\displaystyle a_{N} \leq \sup \{a_{n}: n \geq N \} $ and $\displaystyle b_{N} \geq \inf \{b_{n}: n \geq N \} $.

Now what?