MathDB

The number

x

from

[0,1]

is written as an infinite decimal fraction. Having rearranged its first five digits after the point we can obtain another fraction that corresponds to the number

x_1

. Having rearranged five digits of

x_k

from

(k+1)

-th till

(k+5)

-th after the point we obtain the number

x_{k+1}

.
a) Prove that the sequence

x_i

has limit.
b) Can this limit be irrational if we have started with the rational number?
c) Invent such a number, that always produces irrational numbers, no matter what digits were transposed.

Problem Statement