An Infimum of Superior Limits

  • #1
Euge
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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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  • #2
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.
 
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