assume x is the largest real number less than 1
so the average of x and 1 (1+x)/2 is greater than x but less than 1
this contradictions with the assumption
so there isn't a largest real number less than 1
You can't do that. I think you misunderstood me; while I understand what you are saying, it cannot be done. How do you make x = 0.9 repeated in any actual real life situation, in any form? This is like saying that you cannot have infinity because infinity +1 is bigger... it doesn't really work.