# Tagged Questions

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### Suppose that for $a_n\geq b_n$ for all $n$. Show that $\varliminf_{n \to \infty} a_n\geq \varliminf_{n \to \infty} b_n$.

This is what I have so far: Since $a_n\ge b_n$ for every $n$ then we have that $\inf\{a_n; n\ge k\} \ge \inf\{b_n; n\ge k\}$ for every $n$. When we take the limit as $n\rightarrow \infty$ we get ...
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Sometimes, when we take limits, especially for roots and ratio tests, we define lim of sup(a_n). Is this only because when we take the limit of sup we don't have to worry about the existence of the ...
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### Can we prove $\displaystyle \limsup_{n \to \infty} \sin(n) = 1$?

Can we prove that $\displaystyle \limsup_{n\to \infty} \sin(n) = 1$? I can prove that the above statement holds assuming that $\displaystyle \frac{\pi}{2}$ is normal (this fact is used somewhat ...
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### Limit superior and inferior

How can I find the limit superior/inferior of $a_n$, as $n \rightarrow \infty$? $$a_n=\frac{n^2+4n-5}{n^2+9}\sin^2\left(\frac{n\pi}{4}\right), n \in \mathbb N$$ I've tried Wolfram|Alpha, but it ...
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### Prove that $\liminf x_n = -\limsup (-x_n)$

How can we prove that $\liminf x_n = -\limsup (-x_n)$?
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### $\limsup c_n\le \max(\limsup a_n,\limsup b_n)$

have a question that im stuck on here Let $a_n, b_n$ and $c_n$ be three sequences of real numbers. Suppose $k_n \in [0,1]$ for all $n$. Let $$c_n = (k_n)(a_n) + (1-k_n)b_n\;.$$ Assuming that ...
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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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### Probability measures on $\mathbb{T}$ whose Fourier coefficients tend to 1

Let $\mu$ be a probability measure on the complex unit circle $\mathbb{T}$. Are the following two assertions equivalent? $\limsup_{n\to\infty}|\hat{\mu}(n)|=1$. There exists an increasing sequence ...
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### Proving that $\limsup a_n$ and $\liminf a_n$ are subsequential limits of $\{a_n\}$

Prove that if $\{a_n\}$ is a sequence, then $\limsup a_n$ and $\liminf a_n$ are subsequential limits of $\{a_n\}$. I don't know the case where $\limsup a_n = \infty$.
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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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### 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 ...
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### 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 ...
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### Limit superior and inferior reference request.

Is there any book where I can find theory on the limit superior and limit inferior, plus a nice deal of excercises in the same spirit as Spivak's type of excercises?
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### How to Quantify $\limsup \limits_{i \to \infty} \; E_i = \bigcap \limits_{k=1}^{\infty} \bigcup \limits_{i=k}^{\infty}\; E_i$?

$$\limsup_{i \to \infty} \; E_i = \bigcap_{k=1}^{\infty} \; \bigcup_{i=k}^{\infty} \; E_i$$ So $$\bigcup_{i=k}^{\infty} \; E_{i} = \{x \in E_{i} \mid i \in I\}= S$$ where $I$ is some index set. When ...
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### Proof: Limit superior intersection

How to prove $\limsup(\{A_n \cup B_n\}) = \limsup(\{A_n\}) \cup \limsup(\{B_n\})$? Thanks!
A sequence of sets is defined as $A_n=\{x \in [0,1] : |\sum_{i=0}^{n-1} 1_{[\frac{i}{2n},\frac{2i+1}{4n})} - 1_{[\frac{2i+1}{4n},\frac{i+1}{2n})}| \geq p\}$ for some positive $p\geq0$. What is ...
I am looking for an intuitive explanation of $\liminf$ and $\limsup$ for sequence of sets and how it corresponds to $\liminf$ and $\limsup$ for sets of real numbers. I researched online but cannot ...