IIRC the first number proven to be transcendental was Liouville's number 0.110001000000000000000001... where the digits after the decimal place are 1 in the n

IIRC the first number proven to be transcendental was Liouville's number 0.110001000000000000000001... where the digits after the decimal place are 1 in the n!th place and 0 otherwise. This is explicitly constructed to be very close to the sequence of rational numbers 0.1, 0.11, 0.110001, ... (I expect there's nothing special about base 10 here; surely the proof works in binary as well.)