Question 884187: Prove there is no largest real number in ~R where ~R is the complement of R the real number set.
Found 2 solutions by fcabanski, jack.wang5:
Answer by fcabanski(1391)   (Show Source):
You can put this solution on YOUR website! One way to prove this is by contradiction.
Assume n is the largest real number. Multiply n by 2. 2n is a larger real number than n, so n cannot be the largest real number.
Answer by jack.wang5(5) (Show Source):
You can put this solution on YOUR website! By the Cantor–Bernstein–Schroeder theorem that c = 2^|N|, where c is the cardinality of the real number set, |N| cardinality of the natural number set, this implies that, given c, there exists a real number R|c being, namely, the largest real number in R.
However, since (~R and R) is empty, the R|c does not exist in ~R. Therefore there is no largest real number in ~R.
Reference:
Cardinality of the continuum, http://en.wikipedia.org/wiki/Cardinality_of_the_continuum
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