Question

A safe is connected to a quantum supercomputer, which stores an infinite list of passwords A = {a1,a2,...}. Each password is a real number in the interval [0, 1].

Entering one of the numbers from A will open the safe. The verification process works like this. Whenever someome enters a password number x, the computer takes the first n digits of x and checks if there is a number in A with the same first n digits. If there is a match, it takes the first n + 1 digits of x and checks if there is a number in A with the same first n+1 digits. If it can find a match for all n = 1,2,... (the computer can do infinitely many comparisons in finite time), then it opens the door of the safe. For example, if someone enters π, the computer will check if there is a number in A with the first digit 3, then with the first two digits 3.1, then with the first three digits 3.14 and so on.

a) Is there, necessarily, the smallest password number in the list A?

b) Can it happen that someone enters a number that is not contained in A, but the safe still opens?

c) Can it happen that every number x ∈ [0, 10] will open the safe?

Answer 1