If n is a natural number >= 3, I think your proof is logically acceptable, but its format may not be. That depends on how formal your proofs must be.

A more formal proof might look like:

Assume there exists a natural number k such that k >= 3 and there exists a pair of natural numbers, a[sub:2hgzwvdo]k[/sub:2hgzwvdo] and b[sub:2hgzwvdo]k[/sub:2hgzwvdo], such that (a[sub:2hgzwvdo]k[/sub:2hgzwvdo] + b[sub:2hgzwvdo]k[/sub:2hgzwvdo] + 1) = k.

Let a[sub:2hgzwvdo]k+1[/sub:2hgzwvdo] = a[sub:2hgzwvdo]k[/sub:2hgzwvdo].

So, a[sub:2hgzwvdo]k+1[/sub:2hgzwvdo] is a natural number.

Let b[sub:2hgzwvdo]k+1[/sub:2hgzwvdo] = b[sub:2hgzwvdo]k[/sub:2hgzwvdo] + 1.

So, b[sub:2hgzwvdo]k+1[/sub:2hgzwvdo] is a natural number.

(a[sub:2hgzwvdo]k+1[/sub:2hgzwvdo] + b[sub:2hgzwvdo]k+1[/sub:2hgzwvdo] + 1) = [a[sub:2hgzwvdo]k[/sub:2hgzwvdo] + (b[sub:2hgzwvdo]k[/sub:2hgzwvdo] + 1) + 1] = [(a[sub:2hgzwvdo]k[/sub:2hgzwvdo] + b[sub:2hgzwvdo]k[/sub:2hgzwvdo] + 1) + 1] = (k + 1).

And (k + 1) is a natural number >= 3.

So, (k + 1) is a natural number >= 3 such that there exists a pair of natural numbers, a[sub:2hgzwvdo]k+1[/sub:2hgzwvdo] and b[sub:2hgzwvdo]k+1[/sub:2hgzwvdo], such that (a[sub:2hgzwvdo]k+1[/sub:2hgzwvdo] + b[sub:2hgzwvdo]k+1[/sub:2hgzwvdo] + 1) = (k + 1)

If there exists a natural number k such that k >= 3 and there exists a pair of natural numbers, a[sub:2hgzwvdo]k[/sub:2hgzwvdo] and b[sub:2hgzwvdo]k[/sub:2hgzwvdo], such that (a[sub:2hgzwvdo]k[/sub:2hgzwvdo] + b[sub:2hgzwvdo]k[/sub:2hgzwvdo] + 1) = k,

then, for every natural number n >= k, there exists a pair of natural numbers, a[sub:2hgzwvdo]n[/sub:2hgzwvdo] and b[sub:2hgzwvdo]n[/sub:2hgzwvdo], such that a[sub:2hgzwvdo]n[/sub:2hgzwvdo] + b[sub:2hgzwvdo]n[/sub:2hgzwvdo] + 1 = n.

Let a[sub:2hgzwvdo]3[/sub:2hgzwvdo] = 1, which is a natural number.

Let b[sub:2hgzwvdo]3[/sub:2hgzwvdo] = 1, which is a natural number.

(a[sub:2hgzwvdo]3[/sub:2hgzwvdo] + b[sub:2hgzwvdo]3[/sub:2hgzwvdo] + 1) = 3, which is a natural number >= 3.

THUS, for every natural number n >= 3, there exists a pair of natural numbers, a[sub:2hgzwvdo]n[/sub:2hgzwvdo] and b[sub:2hgzwvdo]n[/sub:2hgzwvdo], such that a[sub:2hgzwvdo]n[/sub:2hgzwvdo] + b[sub:2hgzwvdo]n[/sub:2hgzwvdo] + 1 = n.

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