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.

 

[billtodd]

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

Does a proof by definition of limsup and infimum of a set also work? I mean not through contradiction as you suggest.

For example one can look at the sequence $a_n=n/x$, and see that $((1+a_{n+1})/a_n)^n$ converges to $e^{x+1}$, where $x>0$, since we take the infinimum over all limsup, then when $x\to 0$ we get the infimum here.
This is seen as the optimum.
But I don't see how show that every other positive sequence gives a bigger limsup.