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I would really like help on this probably simple proof:

That the map x |--> f(x) = (x+2)/3 on [0,1] is a contraction,

and maps the ternary Cantor set into itself. Also, find it's fixed point.

(1) I can easily show the fixed point (where f(x) = x) is 1.

(2) I can also pretty easily show it is a contraction:

where |f(x) - xo| <= q*|x - xo|, where q < 1, and xo is the fixed point.

(3) However, I can't seem to find a way to tell whether it maps the ternary Cantor set into itself. I kno the definition of the ternary Cantor set is taking the interval [0,1] and deleting the middle-third of the interval, and then repeating the process on each remaining interval, infinitely.

What does it mean by mapping the ternary Cantor set into itself? Does x have to start out being in the ternary Cantor set? If so, how is it possible if x can vary from [0,1]? What am I interpreting wrong?

Thanks,

Mark