Limit of a sequence in a closed interval is in that interval

In summary, the conversation discusses a closed interval [a,b] on the real numbers and a sequence {xn}n>=1 that belongs to [a,b] and has a limit x as n approaches infinity. The task is to prove that x also belongs to [a,b] using the definition of a limit and Cauchy's theorem. A potential approach is to use proof by contradiction and show that assuming x is outside of [a,b] leads to a contradiction.
  • #1
missavvy
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Homework Statement


Suppose [a,b] is a closed interval (on R), and {xn}n>=1 is a sequence such that
a) xn belongs to [a,b]
b) lim as n--> infinity xn = x exists
prove x belongs to [a,b]


Homework Equations





The Attempt at a Solution



Well since any sequence is bounded, then obviously the limit has to be within the bounds.

Not sure where to begin though. I'm thinking of using the definition of a limit..? For all k>0, there exists a natural # N such that for all n>=N, |x-xn|<k
Or that this is Cauchy since it is bounded ?

Just not exactly how to go about showing that x is in that interval!
 
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  • #2
I would do a proof by contradiction. Assume x is outside of [a,b]. Isn't it pretty easy to derive a contradiction?
 

What is a limit of a sequence in a closed interval?

The limit of a sequence in a closed interval refers to the value that the terms of the sequence approach as the index of the terms increases, while remaining within the given interval.

How is the limit of a sequence in a closed interval calculated?

The limit of a sequence in a closed interval is calculated by evaluating the terms of the sequence as the index approaches infinity, and determining the value that the terms converge to within the given interval.

Why is the limit of a sequence in a closed interval important?

The limit of a sequence in a closed interval is important because it allows us to determine the behavior and convergence of a sequence within a specific range of values. It also helps in understanding the properties and patterns of the sequence.

Can the limit of a sequence in a closed interval be outside of the interval?

No, the limit of a sequence in a closed interval must always be within the given interval. This is because the terms of the sequence are bounded by the interval, and the limit is the value that the terms approach within that bound.

What is the difference between a limit of a sequence in a closed interval and a limit of a sequence in an open interval?

The main difference is that the limit of a sequence in a closed interval must be within the given interval, while the limit of a sequence in an open interval can be outside of the interval. This is because an open interval does not have a defined boundary for the terms of the sequence.

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