I need help with writing a proof about cardinality. Please look at description for more detail.

Question

Let N^N(Nautral Numbers)={f:f|N->N}.

 

a. Give two elements of N^N

 

b. Prove that Aleph Zero <|N^N|

Answer 1

The proof uses the famous Cantor's diagonal argument which can be found in many different places online. Unfortunately, it can not be written here. The idea is to assume that there is an enumeration of all possible sequences of natural numbers and then find another sequence that it has been "forgotten" from this enumeration. This implies a contradiction and hence, it concludes the argument.