5
votes
3answers
92 views

Proving $\limsup\frac 1 {a_n}=\frac 1 {\liminf a_n}$ and $\limsup a_n\cdot \limsup \frac 1 {a_n} \ge 1$

Let $a_n$ be a sequence such that $\forall n\in \mathbb n: 0<a\le a_n\le b <\infty.$ Prove: $\displaystyle\limsup_{n\to\infty}\frac 1 {a_n}=\frac 1 ...
1
vote
0answers
49 views

Show that $\liminf\limits_{n\rightarrow\infty}a_{n}/a_{n-k}\leq \liminf\limits_{n\rightarrow\infty}a_{n}^{k/n}$

Here is an exercise: Let $\{a_{n}\}$ be a positive increasing sequence, can we prove that: $\liminf_{n\rightarrow\infty}\frac{a_{n}}{a_{n-k}}\leq \liminf_{n\rightarrow\infty}a_{n}^{k/n}$? Could ...
2
votes
1answer
100 views

Proving the Kochen-Stone lemma using the Paley-Zygmund inequality

I am trying to understand a proof to a lemma by Kochen and Stone which appears here, using the Paley-Zygmund inequality. I'll repeat the proof in a detailed manner, and explain what bothers me about ...
2
votes
1answer
64 views

Show that $\lim \inf a_n\le\lim\inf s_n.$

Let $\{a_n\}$ be a bounded sequence of real numbers. Let $s_n=\dfrac{a_1+a_2+\cdots+a_n}{n}~\forall~n\in\mathbb N.$ Show that $\lim \inf a_n\le\lim\inf s_n.$ The only definition I know of limit ...
2
votes
1answer
148 views

Show root test is stronger than ratio test

Let $a_n$ be a sequence of real positive numbers. Show $$\liminf_{n\to  \infty} \frac{a_{n+1}}{a_n} \leq \liminf_{n\to \infty}\,(a_n)^{1/n}  \leq \limsup_{n\to \infty} \, (a_n)^{1/n} \leq ...
5
votes
2answers
78 views

Proving a lower bound on the limit superior of a sequence.

Prove that for every positive sequence {$a_{n}$}, $$\varlimsup_{n \to \infty}\frac{\sum_{i=1}^{n+1}a_{i}}{a_{n}}\geq 4$$ Also find the sequences {$a_{n}$} for which 4 is attained. Attempted ...
3
votes
4answers
159 views

How to prove that $\limsup _{n \rightarrow \infty} f(x_n) \geq f(\limsup _{n \rightarrow \infty} (x_n))$

We were given HW where it is asked to prove that, $\limsup_{n \rightarrow\infty} f(x_n) \geq f(\limsup_{n \rightarrow \infty} (x_n))$, for any bounded sequence $\{x_n\}$ and continuous function ...
1
vote
2answers
135 views

Prove that $\limsup _{n\to \infty}(a_n+b_n)\leq \limsup _{n\to \infty}a_n + \limsup _{n\to \infty}b_n$

Let $(a_n)$ and $b_n$ be bounded sequences of real numbers. Prove that $$\limsup _{n\to \infty}(a_n+b_n)\leq \limsup _{n\to \infty}a_n + \limsup _{n\to \infty}b_n$$ How can this be proved? Using the ...
5
votes
2answers
204 views

Limit superior inequalities proof

Let $a_n$ be a positive sequence. Prove that $$\limsup_{n\to \infty} \left(\frac{a_1+a_{n+1}}{a_n}\right)^n\geqslant e.$$
1
vote
3answers
257 views

Proof for limit superior's property: $\limsup (a_n b_n ) \leq \limsup a_n \cdot \limsup b_n$ [duplicate]

Let $a_n,b_n>0$ for all $n\in\mathbb N$. Prove that $\limsup (a_n b_n ) \leq \limsup a_n \cdot \limsup b_n$ I know that $\limsup (a_n+b_n ) \leq \limsup a_n + \limsup b_n$. But I don't know how ...
0
votes
1answer
169 views

show that $ \liminf A \lt \liminf B$

This is Exercise 12.12 in the Ross textbook and I found the solution like the below, but I'm not sure how the yellow line is resulted from the above lines. Can you help me understand this solution ...
1
vote
1answer
284 views

Proving a lim sup inequality in Chapter 12 of Kenneth Ross's “Elementary Analysis”

I am struggling to digest and understand Theorem 12.2 (p. 76) in Elementary Analysis: The Theory of Calculus by Kenneth Ross Theorem 12.2 Let $s_n$ be any sequence of nonzero real numbers. ...
1
vote
1answer
326 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 ...
3
votes
2answers
159 views

Inequality between limits inferior [duplicate]

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
149 views

Proof of limit inequality [duplicate]

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
108 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 ...
1
vote
2answers
172 views

Average limit superior [duplicate]

Let $\mathcal{l}_\mathbb{R}^\infty$ be the space of bounded sequences in $\mathbb{R}$. We define a map $p: \mathcal{l}_\mathbb{R}^\infty\to\mathbb{R}$ by $$p(\underline x)=\limsup_{n\to\infty} ...
0
votes
1answer
127 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
377 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 $\limsup a_n \nleq \limsup b_n$. ...
1
vote
1answer
401 views

How to prove these inequalities: $\liminf(a_n + b_n) \leq \liminf(a_n) + \limsup(b_n) \leq \limsup(a_n + b_n)$

The inequalities are: $$\liminf(a_n + b_n) \leq \liminf(a_n) + \limsup(b_n) \leq \limsup(a_n + b_n)$$
5
votes
0answers
681 views

Proof of $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$

Let $a_n>0$ and $b_n\geq 0$, then $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$ My attempt at a proof is as follows. Let $A_n=\sup\{a_n, a_{n+1},...\}$, $B_n=\sup\{b_n, b_{n+1},...\}$, and ...
6
votes
1answer
293 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
1answer
434 views

Proof that if $s_n \leq t_n$ for $n \geq N$, then $\liminf_{n \rightarrow \infty} s_n \leq \liminf_{n \rightarrow \infty} t_n$

This is half of Theorem 3.19 from Baby Rudin. Rudin claims the proof is trivial. What I've come up with so far doesn't seem trivial, however, and is probably also wrong (my problem with it is pointed ...
4
votes
3answers
2k views

lim sup inequality $\limsup ( a_n b_n ) \leq \limsup a_n \limsup b_n $

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
3k 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
1k 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
755 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 ...