Question

Let X=2N={x=(x1,x2,…):xi∈{0,1}} and define d(x,y)=2∑(i≥1)(3^−i)*|xi−yi|. Define f:X→[0,1] by f(x)=d(0,x), where 0=(0,0,0,…). Prove that maps X onto...

Let X=2N={x=(x1,x2,…):xi∈{0,1}} and define

d(x,y)=2∑(i≥1)(3^−i)*|xi−yi|.

Define f:X→[0,1] by f(x)=d(0,x), where 0=(0,0,0,…).

Prove that maps X onto the Cantor set and satisfies (1/3)*d(x,y)≤|f(x)−f(y)|≤d(x,y) for x,y∈2N.

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