Suppose I draw balls out of a jar and every ball has an infinitely long decimal expansion of a real between 0 and 1. My intent is to show a number that isn't in the jar to the precision of the number of balls drawn out of the jar.
So if the first ball has an 8 in the first position after the decimal point I will write down a 7 in the first position otherwise I will write down an 8 in the first position. I repeat the procedure for the second ball and the second position. I do this over and over.
You say my number is in the set of balls still in the jar and search for one that matches all of the digits I've written down so far and hand it to me. I repeat the same process as before.
Now you might claim my 100 digit number for the first 100 balls isn't the same as the one I might have created had the balls been presented to me in a different order. Well, I don't care. No matter how you shuffle the balls already presented to me, none of them will have the same first 100 digits as the number I generated.