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In the following theorem, why does the limit end up being e from (14) and (15)? I thought we would need to show that the limsup and the liminf are equal to e for the limit to equal e?

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José Carlos Santos
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random_0620
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    You might find this old answer of mine helpful. In that answer, I fill in the missing details of Rudin's rather cryptic proof of this theorem. In particular, see the start of my answer, where I say "the structure of this proof is..." –  Jun 14 '21 at 20:27
  • Many (including myself) take the first line as the definition of $e$. – K.defaoite Jun 14 '21 at 21:17

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Since$$e\leqslant\liminf\nolimits_nt_n\leqslant\limsup\nolimits_nt_n\leqslant e,$$we have$$e=\liminf\nolimits_nt_n=\limsup\nolimits_nt_n,$$which is the same thing as asserting that $e=\lim_nt_n$.

José Carlos Santos
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