Is Every Convergent Sequence of Real Numbers Bounded?

AI Thread Summary
A convergent sequence of real numbers {cn} is indeed bounded, as it approaches a limit L, ensuring that for any chosen epsilon, there exists an N such that for all n greater than N, the terms of the sequence are confined within a specific range around L. This implies that the sequence can only take on values that are ultimately close to L, which allows for the establishment of a bound R that encompasses all terms in the sequence, including the finite initial terms. The discussion emphasizes that even though the sequence may have infinitely many terms, the boundedness still holds due to the finite nature of the initial terms and their relationship to L. Thus, every convergent sequence of real numbers is bounded.
swuster
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Homework Statement


If {cn} is a convergent sequence of real numbers, does there necessarily exist R> 0 such that |cn|≤ R for every n ∈ N? Equivalently, is {cn : n ∈ N} a bounded set of real numbers? Explain why or why not.


Homework Equations


n/a


The Attempt at a Solution


I would think this is patently obvious given the definition of convergence but I don't really know how to put the proof in words and numbers.

cn approaches some value c for large n but it can do so in a number of ways so how can i prove that there is always some R that is larger or equal to all elements cn?

Thanks for the help!
 
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Write the limit definition of a sequence out. Assume to opposite of your claim (for contradiction).
So for any N you pick to satisfy an epsilon of the limit, there will be an R as large as you want where R < |Xn|. Where epsilon is fixed already. So your R can be dependant on epsilon and L. Is there some R that would bring about a contradiction?
 
Suppose your sequence converges to L. Then, by definition of "limit", given any \epsilon&gt; 0 there must exist some N such that if n> N then |a_n- L|&lt; \epsilon.

Take \epsilon= 1, say. Then there exist N such that if n> N, |a_n- L|&lt; 1 which is the same as saying -1&lt; a_n- L&lt; 1 or L-1&lt; a_n&lt; L+ 1. Now that only restricts x_n for n> N, but the set \{a_0, a_1, a_2, \cdot\cdot\cdot, a_N\} is finite and has a largest member. Every number in the sequence \{a_n\} must be less that the that largest member or L+ 1, whichever is larger.
 
In my case {cn} need not be finite, though. Does that still hold in this case?
 
swuster said:
In my case {cn} need not be finite, though. Does that still hold in this case?
Yes. HallsOfIvy is talking about the terms in the sequence c1, c2, ..., cN, the first N terms at the beginning of the sequence.
 
I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
Essentially I just have this problem that I'm stuck on, on a sheet about complex numbers: Show that, for ##|r|<1,## $$1+r\cos(x)+r^2\cos(2x)+r^3\cos(3x)...=\frac{1-r\cos(x)}{1-2r\cos(x)+r^2}$$ My first thought was to express it as a geometric series, where the real part of the sum of the series would be the series you see above: $$1+re^{ix}+r^2e^{2ix}+r^3e^{3ix}...$$ The sum of this series is just: $$\frac{(re^{ix})^n-1}{re^{ix} - 1}$$ I'm having some trouble trying to figure out what to...

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