1
vote
0answers
43 views

Limit superior does not increase when a sequence is replaced by its sequence of averages [duplicate]

Let $\{x_n\}$ be a bounded sequence of real numbers, and define a new sequence $\{\sigma_n\}$ by $$\sigma_n=\frac 1n\sum_{i=1}^nx_i.$$ Prove that $\limsup \sigma_n\le \limsup x_n$. I am confused on ...
1
vote
1answer
60 views

Show that $ \limsup x_n ≤ \limsup y_n$ and $\liminf x_n ≤ \liminf y_n$

Let $(x_n)$ and $(y_n)$ be bounded sequences such that $x_n ≤ y_n$ for all $n \in \mathbb{N}$. Show that $\limsup x_n ≤ \limsup y_n$ and $\liminf x_n ≤ \liminf y_n$.
5
votes
3answers
122 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
50 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
188 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
75 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
196 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
83 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
185 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 ...
2
votes
2answers
188 views

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

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
235 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
281 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
185 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 ...
2
votes
1answer
306 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. ...
2
votes
3answers
325 views

Clarifying a proof of $\limsup (a_n+b_n) \le \limsup a_n + \limsup b_n$

Could you help me understand the solution below? "otherwise we clearly have the equality" -> why? It's not clear to me. :( "The inequality is trivially satisfied" -> why? even if the right side is ...
7
votes
2answers
936 views

Prove $\limsup\limits_{n \to \infty} (a_n+b_n) \le \limsup\limits_{n \to \infty} a_n + \limsup\limits_{n \to \infty} b_n$

I am stuck with the following problem. Prove that $$\limsup_{n \to \infty} (a_n+b_n) \le \limsup_{n \to \infty} a_n + \limsup_{n \to \infty} b_n$$ I was thinking of using the triangle inequality ...
2
votes
2answers
410 views

Inequality concerning limsup and liminf of Cesaro mean of a sequence [duplicate]

Let $\{x_n \}$ be a sequence of real numbers and let $y_n = \frac{(x_1 + x_2 + ... + x_n)}{n}$. (a) Prove that $\liminf x_n \le \liminf y_n \le \limsup y_n \le \limsup x_n$ (b) Give an example of a ...
1
vote
1answer
385 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
165 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
159 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
129 views

Is it true that $\limsup \phi\le\limsup a_n$, where $\phi=\frac{a_1+…+a_n}n$? [duplicate]

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
190 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
134 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
394 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
468 views

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

The inequalities are: $$\liminf(a_n + b_n) \leq \liminf(a_n) + \limsup(b_n) \leq \limsup(a_n + b_n)$$
5
votes
0answers
766 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 ...
7
votes
1answer
364 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: ...
3
votes
2answers
492 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
3k 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
2k 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.
6
votes
1answer
817 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 ...