infinitely descending natural numbers Show that there is no infinitely descending sequence of natural numbers.

I was thinking that there exists no infinite descending chain on the natural numbers, since every chain of natural numbers has a minimal element. And so it reaches a finite minimum.

P.s I am just looking for a well built solution, since i cannot express the proof very clearly
Suppose the first number in the sequence is $n$. Then there can be at most $n$ terms in the sequence, since the second term is at most $n-1$, the third $n-2$, etc.
An integer is finite, and a finite number cannot infinitely decent, by definition