The proof idea is correct, the actual proof has some issues.
Let's go step by step.
First the title:
Is this a valid proof that natural numbers are infinite?
No. There are no valid proofs that natural numbers are infinite, because natural numbers are not infinite. However, fortunately your post does not attempt to try this futility, rather it tries to prove that the set of natural numbers is infinite. This is an important distinction, and if you don't make it, sooner or later you'll make the mistake in a context where it is not obvious that what you say is not what you meant. Just remember that "infinite" is not a number, but a property.
OK, but now to the actual proof:
- Let $\mathbb N$ be the set of natural numbers.
OK, here you just introduce the common notation; of course there's nothing wrong with that.
- Assume that $\mathbb N$ is finite.
So you're going to prove by contradiction. Still OK.
Now consider an arbitrary number $K$, where $K$ is the largest number in $\mathbb N$.
That sentence is self-contradictory. Either $K$ is arbitrary, then it might not be the largest number in $\mathbb N$. Or $K$ is fixed to be the largest number of $\mathbb N$, then it is not arbitrary. From the following, it is obvious that you meant the second.
However, you are making a leap here: To define $K$ as the largest element of $\mathbb N$ you first have to establish that, under the assumption, $\mathbb N$ has a largest element. Now the reason why this is true is that
- $\mathbb N$ is totally ordered
- Every totally ordered finite set has a maximal element.
You have to at least state the reason (e.g. "Be $K$ the largest element of $\mathbb N$; this exists because $\mathbb N$ is totally ordered and by assumption finite). If any of the results is not already known (either as axiom or as previous theorem), you'll also have to prove that.
$K+1$ is also a natural number such that $K+1>K$.
Here you should state based on what that statement is true. For example:
"But if $K$ is a natural number, due to (the axioms or a previously proved theorem) $K+1$ is another natural number with $K+1>K$."
Of course, as before, you can only use that if it is already in the axioms or a previous theorem; otherwise you'll first need to prove it as well.
- Therefore, $\mathbb N$ cannot be finite.
You should explicitly state that the result of point 3 is a contradiction. For example:
"But this means that, in contradiction to the assumption, $K$ cannot be the maximal element. Therefore, $\mathbb N$ cannot have a maximal element, and thus cannot be finite."