In a Combinatorics text, I find this:
Not all infinite sets have the same cardinality. Consider the set of all integers and the set of all reals. Assume that the set of reals can be put in one-to-one-onto correspondence with the integers. Then consider the real number whose ith digit after the decimal is the ith digit of the ith real plus 5 mod 10. This real number cannot be in correspondence with any integer, since it differs from every real that has been mapped to an integer. From this we conclude that the reals cannot be placed in one-to-one correspondence with the integers.
What the bolded text is really saying to prove that one real number has been missed from mapping? Can you please explain the bolded sentence? I want to know how exactly has that real number missed the mapping.