Why the set 'N' has infinite elements

[dbngshuroy]

I dont know where to post this question . So I am posting here.

Why the set of all natural numbers (N) has infinite number of elements. :?:

I have started from this....
Suppose N is finite and its largest element is x. Then x + 1 will be a natural number because addition of two natural num make another natural number which is greater than each previous numbers. clearly x+1>x which is contradicting x is largest. So N has no largest element. so N has infinite number of elements.

another way.......
we can build the set N = { x+1 | x is the element of N }
so if 1 is a natural number then 2 is also . as 2 is the element of N then 3 is also and so on 3,4,5,6,7,8,9,10,..........

So N is infinite.

Is these 2 proof is correct? If there there is any strong proof then kindly post them with some explanation.

 

[pka]

The reason that set of natural numbers, \(\displaystyle \mathbb{N}\), is infinite is bijects with a proper subset of itself.
If \(\displaystyle \mathbb{E}=\{2n:n\in \mathbb{N}\}\) then \(\displaystyle f:n\mapsto 2n\) is a bijection, \(\displaystyle f:\mathbb{N}\to \mathbb{E}\).

 

[lookagain]

I dont know where to post this question . So I am posting here.

Why the set of all natural numbers (N) has infinite number of elements. :?:

There is no last element as you can add 1 to the previous number. The set of
natural numbers is unbounded.