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.

I dont know where to post this. So I am publishing this here.