Prove the existence of uncountable sets using Cantor's diagonalization.

Cant or ' s diagonal ization is a proof technique that can be used to show the existence of un count able sets . It is based on the idea of creating a new sequence that is different from any existing sequence . \n \n Here is an example of how Cant or ' s diagonal ization can be used to prove the existence of an un count able set : \n \n Let ' s consider the set of all integers . We can enumerate this set as follows : \n \n 1 , 2 , 3 , 4 , ... \n \n Now , let ' s consider the set of all even numbers : \n \n 2 , 4 , 6 , 8 , ... \n \n We can see that the even numbers can be listed in a one - to - one correspond ence with the integers , since every even number can be pa ired with its corresponding odd number . \n \n However , Cant or ' s diagonal ization allows us to create a new even number that is different from any existing even number . Here ' s how : \n \n Let ' s consider the number 2 . We can create a new even number by adding a small amount to 2 : \n \n 2 + 1 = 3 \n \n Now , let ' s consider the number 3 . We can create a new even number by adding a small amount to 3 : \n \n 3 + 1 = 4 \n \n And so on . By adding a small amount to the previous even number , we create a new even number that is different from any existing even number . \n \n This process can be repeated in def initely , creating an infinite number of new even numbers that are different from any existing even number . Since there are more even numbers than there are integers , this means that the set of even numbers is un count able . \n \n C ant or ' s diagonal ization can be used to prove the existence of other un count able sets as well , by creating new sequences that are different from any existing sequences . </s>