Before starting reading this note please go through my note on uncountability of . There I have shown that , the set of real numbers, is uncountable. Now in this note, I will show a that there exists a bijection between and . Well, obviously such bijection doesn't exists and my method has a fallacy, but I want you people to find out the fallacy yourself.
I will construct a function as follows:
From choose an arbitrary element and map it to an arbitrary element from .
Next choose an arbitrary element from and map it to arbitrary element of and continue the process.
Thus in the i-th step map the arbitrary element to an arbitrary element .
Then, from the construction of , is injective. Choose any two different elements from , and observe that they are mapped to two elements of under , which must be different (different because whenever in a step an element of is mapped to that of , the latter is deleted from before proceeding to the next step). Again is surjective because choose any arbitrary element from , see that it is the image of an element of (Since our process of construction of is infinite, this means at some step of our process we must have mapped an element of to that element of ).
So turns out to be bijective. Clearly, there is a strong fallacy in my considerations. Can anybody point it out?