What Is the Infimum of This Set of Superior Limits?

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Let $\mathbb{R}^\omega_{+}$ be the set of all sequences $(a_n)_{n=1}^\infty$ of positive real numbers. Determine the infimum $$\inf\left\{\limsup_{n \to \infty} \left(\frac{1 + a_{n+1}}{a_n}\right)^n : (a_n) \in \mathbb{R}^\omega_{+}\right\}$$

 

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Spoiler: Hint

First show, by way of contradiction, that $e$ is a lower bound of the set of superior limits. Then show that $e$ is the optimal bound.