Is it possible to construct a sequence $a_n$ satisfying
(1) $\forall n,a_n>0$
(2) $\limsup_{n\to \infty}\left(\frac{1+a_{n+1}}{a_n}\right)^n=e$
If I use the reccurance $a_n+1=(1+\frac{1}{n})a_n-1$ for all terms, the sequence will become negative after some steps, while if the reccurance holds for some term, the upper bound will be greater than $e$, so I'm confused whether the sequence exists or not.
Thank you for your attention!