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

Click For Summary
SUMMARY

The limit of a sequence {xn} that is contained within a closed interval [a,b] converges to a point x that also lies within the interval [a,b]. This conclusion is derived from the properties of bounded sequences and the definition of limits. Specifically, since the sequence is bounded, the limit must also fall within the bounds of the interval. A proof by contradiction can be effectively employed to demonstrate that assuming x is outside [a,b] leads to a contradiction.

PREREQUISITES
  • Understanding of closed intervals in real analysis
  • Familiarity with the definition of limits
  • Knowledge of bounded sequences
  • Experience with proof techniques, particularly proof by contradiction
NEXT STEPS
  • Study the properties of closed intervals in real analysis
  • Learn about the formal definition of limits in calculus
  • Explore the concept of bounded sequences and their implications
  • Practice proof by contradiction with various mathematical statements
USEFUL FOR

Students of real analysis, mathematicians, and educators looking to deepen their understanding of limits and sequences within closed intervals.

missavvy
Messages
73
Reaction score
0

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!
 
Physics news on Phys.org
I would do a proof by contradiction. Assume x is outside of [a,b]. Isn't it pretty easy to derive a contradiction?
 

Similar threads

Prove a theorem about a vector space and convex sets
  • · Replies 1 ·
Replies
1
Views
2K
Interval of convergence of power series from ratio test
  • · Replies 2 ·
Replies
2
Views
2K
Bounded non-decreasing sequence is convergent
  • · Replies 5 ·
Replies
5
Views
2K
How to show that these sequences are summable?
  • · Replies 1 ·
Replies
1
Views
2K
Prove: Limit Point of H ∪ K if p is Limit Point of H or K
  • · Replies 5 ·
Replies
5
Views
2K
Intro Real Analysis: Closed and Open sets Of R. Help with Problem
  • · Replies 8 ·
Replies
8
Views
2K
Proof that two equivalent sequences are both Cauchy sequences
  • · Replies 8 ·
Replies
8
Views
3K
How can I show that the sup(S)=lim{Xn} and the inf(S)=lim{Yn} as n goes to infinity
  • · Replies 1 ·
Replies
1
Views
3K
Understanding the Use of Min in Cauchy Sequences
  • · Replies 1 ·
Replies
1
Views
1K
Regarding Real numbers as limits of Cauchy sequences
  • · Replies 28 ·
Replies
28
Views
3K