I want to prove that $\limsup (\frac{1+a_{n+1}}{a_n})^n\ge e$ for any sequence $\{a_n\}$ of positive numbers and the bound cannot be improved. There are different proofs but mine is more direct and elementary.
My solution: if there is a nondecreasing subsequence of $\{a_n\}$, then $(\frac{1+a_{n+1}}{a_n})^n$ contains a subsequence whose terms are greater than or equal to the corresponding subsequence of $\{(1+\frac{1}{n})^n\}$. Therefore the relation holds true in this case.
Otherwise, from some term on the sequence is nonincreasing and therefore the superior limit will equal positive infinity. So the relation holds true in this case as well.
Is my solution free of error?
But I’m not sure how to show that the bound can’t be improved. Any suggestions?