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math104-s22:notes:lecture_4 [2022/01/27 09:22] pzhou |
math104-s22:notes:lecture_4 [2022/01/27 10:46] (current) pzhou [Lecture 4] |
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* Cauchy sequence. | * Cauchy sequence. | ||
- | Discussion time: Ex 10.1, 10.6 in Ross | + | Discussion time: Ex 10.1, 10.7, 10.8 in Ross |
==== limit goes to $+\infty$? ==== | ==== limit goes to $+\infty$? ==== | ||
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==== $\liminf$ and $\limsup$ ==== | ==== $\liminf$ and $\limsup$ ==== | ||
- | Recall the definition of $\sup$. | + | Recall the definition of $\sup$. |
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+ | Also, for a sequence $(a_n)_{n=m}^\infty$, | ||
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+ | Also, for a sequence $(a_n)_{n=1}^\infty$, | ||
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+ | We want to define a gadget, that captures the 'upper envelope' | ||
+ | $$ A_m = \sup_{n \geq m} a_n $$ | ||
+ | then we define | ||
+ | $$ \limsup a_n = \lim A_m (= \inf A_m) $$ | ||
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+ | Time for some examples, $a_n = (-1)^n (1/n)$. | ||
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