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So evidently, a hypothesis of distributing the limit is that we know ##a_n## and ##b_n## converge.

So, here is my question. Say that I don't know whether ##1/n## and ##1/n^2## converges or not. Normally, to evaluate ##\lim (1/n + 1/n^2)## we distribute the limit and then determine whether each sequence converges or not. Shouldn't this be the other way around? Shouldn't we determine whether each sequence converges first, and then distribute the limit, which is what models the logical progression of the theorem above?