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mahler1
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Homework Statement .
Let ##\{I_n\}_{n \in \mathbb N}## be a sequence of closed nested intervals and for each ##n \in \mathbb N## let ##\alpha_n## be the length of ##I_n##.
Prove that ##lim_{n \to \infty}\alpha_n## exists and prove that if ##L=lim_{n \to \infty}\alpha_n>0##, then ##\bigcap_{n \in \mathbb N} I_n## is a closed interval of length ##L##.
The attempt at a solution.
I didn't have problems to prove the existence of the limit: if ##I_n=[a_n,b_n]##, then ##\alpha_n=b_n-a_n## and ##\{a_n\}_n, \{b_n\}_n## are increasing and decreasing bounded sequences respectively, so both are convergent ##\implies \{\alpha_n\}_n## is convergent and ##lim_{n \to \infty} \alpha_n=lim_{n \to \infty}b_n-a_n=lim_{n \to \infty}b_n-lim_{n \to \infty}a_n=b-a##.
Now, I would like to say that if ##b-a>0 \implies \bigcap_{n \in \mathbb N} I_n=[a,b]##, but I don't know hot to prove this part.
Let ##\{I_n\}_{n \in \mathbb N}## be a sequence of closed nested intervals and for each ##n \in \mathbb N## let ##\alpha_n## be the length of ##I_n##.
Prove that ##lim_{n \to \infty}\alpha_n## exists and prove that if ##L=lim_{n \to \infty}\alpha_n>0##, then ##\bigcap_{n \in \mathbb N} I_n## is a closed interval of length ##L##.
The attempt at a solution.
I didn't have problems to prove the existence of the limit: if ##I_n=[a_n,b_n]##, then ##\alpha_n=b_n-a_n## and ##\{a_n\}_n, \{b_n\}_n## are increasing and decreasing bounded sequences respectively, so both are convergent ##\implies \{\alpha_n\}_n## is convergent and ##lim_{n \to \infty} \alpha_n=lim_{n \to \infty}b_n-a_n=lim_{n \to \infty}b_n-lim_{n \to \infty}a_n=b-a##.
Now, I would like to say that if ##b-a>0 \implies \bigcap_{n \in \mathbb N} I_n=[a,b]##, but I don't know hot to prove this part.