What Is the Infimum of This Set 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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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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Euge said:
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.
 
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