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

[missavvy]

Homework Statement

Suppose [a,b] is a closed interval (on R), and {x_n}n>=1 is a sequence such that
a) x_n belongs to [a,b]
b) lim as n--> infinity x_n = 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-x_n|<k
Or that this is Cauchy since it is bounded ?

Just not exactly how to go about showing that x is in that interval!

 

[Dick]

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