TITLE: Limit superior inequalities proof: $\limsup_{n\to \infty} \left(\frac{a_1+a_{n+1}}{a_n}\right)^n\ge e$
QUESTION [7 upvotes]: Let $a_n$ be a positive sequence. Prove that 
$$\limsup_{n\to \infty} \left(\frac{a_1+a_{n+1}}{a_n}\right)^n\geqslant e.$$

REPLY [2 votes]: Since I solved this problem several years ago, I didn't write my solution immediately, so that others could think on this problem. Now I am writing my own solution:
It starts as the solution by Ju'x, i.e. we can safely assume that $a_1 = 1$ and suppose the converse inequality. Then there exists $N \in \mathbb{N}$ such that 
$$\frac{1+a_{n+1}}{a_n} < e^{1/n}, \qquad n \ge N.$$
Hence
$$a_N > \frac{1}{e^{1/N}} + \frac{a_{N+1}}{e^{1/N}} > \frac{1}{e^{1/N}} + \frac{1}{e^{(1/N)+(1/N+1)}} + \frac{a_{N+2}}{e^{(1/N)+(1/N+1)}} > \ldots,$$
i.e.
$$a_N > \frac{1}{e^{1/N}} + \frac{1}{e^{(1/N)+(1/N+1)}} + \ldots + \frac{1}{e^{(1/N)+\ldots+(1/N+k)}}, \qquad k \in \mathbb{N}.$$
Using $e^{1/n} < 1 + \dfrac{1}{n-1} = \dfrac{n}{n-1}$ we get
$$a_N > (N-1)\left( \frac{1}{N} + \frac{1}{N+1} + \ldots + \frac{1}{N+k}\right), \qquad k \in \mathbb{N},$$
which is impossible, since the harmonic series diverges.