Sorry, tried to edit my previous post, but I guess it's still being reviewed.

I'm not sure how to show this clearly because there's alot of "choosing" involved.

If we start with a single digit number, then the case is trivial, because the next number will be twice that single digit (i.e. 2 times that number).

If we start with a multiple digit number, and we try to keep the integers to be odd (by adding an appropriate digit to

), what will happen is that we will end up with an odd integer with all digits odd, so we are forced to add two odd numbers, which results in an even number.