Question

Theorem: there exists a bijection between the Cantor set C and the unit interval [0,1].

Base Case: prove that for ,

·         x₁ = 0 implies x in [1, 1/3],

·         x₁ = 1 implies x in (1/3, 2/3), and

·         x₁ = 2 implies x in [2/3 , 1] .

(must deal with endpoint confusion).

Induction step: At step n, suppose that the ternary expansion

                x = x₁x₂ x₃ … x_n …

comprised ONLY of 0’s and 2’s describes a pathway that leads to a unique closed interval which is still intact. Then,

·         x_n+1 = 0 implies we will be in the left-most third,

·         x_n+1 = 1 implies we will be in the middle third, and

·         x_n+1 = 2 implies we will be in the right-most third.

Must be an Induction proof

Answer 1