I feel like I'm missing something obvious, but anyway, in the text it states:(adsbygoogle = window.adsbygoogle || []).push({});

lim as n→∞ of a_{n}+b_{n}= ( lim as n→∞ of a_{n}) + ( lim as n→∞ of b_{n})

But say a_{n}is 1/n and b_{n}is n. Then the limit of the sum is n/n = 1, but the lim as n→∞ of b_{n}doesn't exist and this property doesn't work...

Second thing, I was reading an example of how to prove a sequence is divergent, specifically the sequence [(-1)^{n}], and the text proves it by contradiction. "Take any positive number [itex]\epsilon[/itex]<1. Then the interval (a - [itex]\epsilon[/itex], a + [itex]\epsilon[/itex]) has length less than 2. Therefore, it is not possible to have both odd and even terms of sequence in this interval, therefore the sequence [(-1)^{n}] cannot converge to a."

Why does not being able to possess both odd and even terms in that interval mean that the sequence doesn't converge? It seems like a random leap of logic to me...

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# Convergent sequence property and proving divergence

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