Proof that lim sup cos(n) is 1

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In summary, the homework statement states that the lim sup cos(n)=1, and by continuity, this implies that |P_n-Q_n\pi|<1/Q_n.
  • #1
tt2348
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Homework Statement



Prove that the lim sup cos(n)=1

Homework Equations



given any [itex]α\in\mathbb{R}-\mathbb{Q} , \exists P_n,Q_n\in\mathbb{N}.\ni.\frac{P_n}{Q_n} \rightarrow {a} [/itex] and by definition, [itex] \frac{1}{Q_n} \rightarrow 0[/itex]

Also, [itex] cos^{2}(x)=\frac{1+cos(2x)}{2}[/itex]

My attempt

I first consider [itex]lim\:sup\:cos^{2}(n) [/itex]
We know that [itex]\exists P_n,Q_n\in\mathbb{N}.\ni.\frac{P_n}{Q_n} \rightarrow {\pi}[/itex] which means [itex]\forall \epsilon>0 \: \exists{N} \: .\ni. \forall{n}>N, \: |\frac{P_n}{Q_n}-\pi|<\epsilon[/itex]
pick [itex]N.\ni.\:\epsilon<\frac{1}{Q_{n}^2}[/itex]
So it follows that [itex]|\frac{P_n}{Q_n}-\pi|<\frac{1}{Q_{n}^2}[/itex]
and therefore [itex]|P_n-Q_n\pi|<\frac{1}{Q_n} [/itex], am I allowed to make assume by continuity that [itex]|cos^{2}(P_n)-cos^{2}(Q_n\pi)|=|cos^{2}(P_n)-1|<ε [/itex] and since [itex] P_k = a_{n_{k}} [/itex] where [itex]a_n=n[/itex]?
 
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  • #2
Why would you do all of that? You know that [itex]cos(x)\le 1[/itex] for all x and that cos(0)= 1. Therefore the maximum value of cos(x) is 1 and so is "lim sup".

(Strictly speaking "lim sup" is defined for sets, not functions, but I assume that by "lim sup cos(x)" you mean "lim sup {y| y= cos(x) for some x}".)
 
  • #3
It's not for a function... but the sequence cos(n)
where n=1,2,3... and so on
obviously I know that cosine is max at 1, but i have to prove that the lim sup is 1 over natural numbers
 
  • #4
to clarify, I need to show that
[itex] \stackrel{limit}{N \rightarrow{\infty}}\:sup\left\{cos(n):n\in\mathbb{N} \wedge n>N\right\}=1[/itex]
 
  • #5
also... having a maximum value of 1 at 0 is not the same as having the lim sup being 1...
max value of 1/n is 1 (n=1)... but the lim sup {1/n:n is a natural number and n>N}=0
 
  • #6
Yes, now that I see that it is the limit of hte sequence, not the function, you are correct.
 
  • #7
but as far as my proof, is it a correct assumption?
 

1. What is the definition of lim sup?

The limit superior, or lim sup, is the largest limit point of a sequence or a function. It represents the largest value that a sequence or function approaches as its input approaches a certain value.

2. What is the significance of cos(n) in this proof?

Cos(n) represents the cosine function evaluated at the natural numbers. In this proof, it is used to show that the sequence cos(n) has a limit superior of 1.

3. How is the proof that lim sup cos(n) is 1 different from other limit proofs?

The proof for lim sup cos(n) is different from other limit proofs because it involves using the properties of the lim sup, such as the fact that it is the largest limit point, rather than direct calculations or algebraic manipulation.

4. Can you provide an example of how this proof can be applied in real-life scenarios?

This proof can be applied in real-life scenarios where sequences or functions are used to model natural phenomena. For example, it can be used to show that the maximum temperature of a system is 1 degree Celsius, even if the temperature fluctuates over time.

5. What are the practical implications of this proof?

The proof that lim sup cos(n) is 1 has practical implications in various fields, including mathematics, physics, and engineering. It can be used to prove the existence of certain limits and to analyze the behavior of sequences and functions in various applications.

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