Tagged Questions
1
vote
1answer
73 views
Prove two inequalities about limit inferior and limit superior
I wish to prove the following two inequalities:
Suppose $X$ is a subset in $\Bbb R$, and functions $f$ and $g$: $X\to \Bbb R$, and $x_{0}\in X$ is a limit point. Then: $$\lim\sup_{x\to ...
4
votes
2answers
89 views
Inequality between limits inferior
Let $(a_n)$ be a non-zero real sequence with $\left (\frac{a_{n+1}}{a_n} \right)$ bounded. How might we prove that
$$\liminf_{n\to\infty} \, \frac{|a_{n+1}|}{|a_n|} \leq \liminf_{n\to\infty} \, ...
1
vote
1answer
75 views
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 $A = \lim\text{sup}\frac{x_{n+1}}{x_n}$. ...
3
votes
1answer
73 views
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 ...
0
votes
1answer
99 views
$\limsup(a\cdot a_n)=a\cdot \limsup(a_n)$
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:
...
1
vote
1answer
262 views
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 $\limsup a_n \nleq \limsup b_n$.
...
0
votes
1answer
185 views
How to prove these inequalities in real analysis?
The inequalities are:
$$\liminf(a_n + b_n) \leq \liminf(a_n) + \limsup(b_n) \leq \limsup(a_n + b_n)$$
6
votes
1answer
156 views
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: ...
2
votes
3answers
980 views
lim sup a simple inequality
I´m not sure how to start with this proof, how can I do it?
$$
\limsup ( a_n b_n ) \leqslant \limsup a_n \limsup b_n
$$
I also have to prove, if $ \lim a_n $ exists then:
$$
\limsup ( a_n b_n ) = ...
4
votes
3answers
1k views
Properties of $\liminf$ and $\limsup$ of sum of sequences
Let $\{s_n\}$ and $\{t_n\}$ be sequences.
I've noticed this inequality in a few analysis textbooks that I have come across, so I've started to think this can't be a typo:
$\limsup\limits_{n ...
3
votes
2answers
758 views
Subadditivity of the limit superior
$$ \limsup \left(f(h)+g(h)\right) \leq \limsup f(h)+ \limsup g(h).$$
How can we prove this? Any help would be appreciated.
5
votes
1answer
452 views
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 ...
