# Tagged Questions

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### Proof of limit inequality

Prove that for any sequence $\{x_n\}$ of positive real numbers $$\lim\text{sup}\sqrt[n]{x_n}\leq \lim\text{sup}\frac{x_{n+1}}{x_n}.$$ My attempt: Let $A = \lim\text{sup}\frac{x_{n+1}}{x_n}$. ...
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### Is it true that $\limsup \phi\le\limsup a_n?$

Define $\phi={a_1+...+a_n\over n}$ and $(a_n)$ be a sequence of real number. Is it true that $\limsup \phi\le\limsup a_n?$ Intuitively i am wondering why the inequality holds as the sup of a sequence ...
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### $\limsup(a\cdot a_n)=a\cdot \limsup(a_n)$

I know it's quite obvious that $\limsup(a\cdot a_n)=a\cdot \limsup(a_n)$ for $a$ a real number >0, but I don't know how to prove it. My second question is whether the following proof works for: ...
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### Prove: $a_n \leq b_n \implies \limsup a_n \leq \limsup b_n$

Is my proof correct? Prove: $a_n \leq b_n \implies \limsup a_n \leq \limsup b_n$ Proof: Let $a_n$ and $b_n$ be sequences such that $a_n \leq b_n \forall_n$. Suppose $\limsup a_n \nleq \limsup b_n$. ...
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### How to prove these inequalities in real analysis?

The inequalities are: $$\liminf(a_n + b_n) \leq \liminf(a_n) + \limsup(b_n) \leq \limsup(a_n + b_n)$$
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### If $\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$ then $\operatorname{{lim sup}}\sigma_n \leq \operatorname{lim sup} s_n$

This is a question from the book Methods of Real Analysis by R. R. Goldberg. If $(s_n)$ is a sequence of real numbers and if $$\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$$ then prove that: ...
### Inequality involving $\limsup$ and $\liminf$
This may have been asked before, however I was unable to find any duplicate. This comes from pg. 52 of "Mathematical Analysis: An Introduction" by Browder. Problem 14: If $(a_n)$ is a sequence in ...