Question

Use Cantor Diagonal Argument to prove that the set {?∈ℝ|9≤?<10} is uncountable infinite.

Answer 1

thal Use cantor Diagonalisation argument to prove the set [NER: 92 x < 10} is uncountable infinite suppose a bijection fin son Arguing thy contradiction, L9, 10) exists Listing tuminating bi-infinite array: the fend by their decimal expansions, we build a f) 0.01942 a 3 anu ais- O. 021 922 923 924 azs 2) = 112) $13) 3) - o. a3,932 933 934 935- - o au, auz auz 314) $15) = any aus.. o asi asa asa asu ass- brinen the anay we can ral no. no explicitly exhibit a NEL210) that it can't possibley include with non. Namely, et n be the terminating decimal expansion: n=o.d, da dz dyds where the an are defined defined using the diagonal entries of the away, modified as follows: dn= anmt I if annt 10, 1, 2....8}; if annia Note that ar to for to this terminating decimal expansion of allowed kind and defines that for all nein, fino&x, contradicting fact t that f is dni 8 if all n non a real no in [a, 10). we claim the onto.

in of To see this, observe that the nth. diget the decimal expansion a is an and in the expansion of fens is don; there are different This concludes the proof. .. the det {ntri qsullo] is uncountable infinite