I wrote a proof by contradiction for the existence of a prime decomposition for all positive integers, I came up with it while I was trying solidify my understanding of the theorem itself. (Just existence, not uniqueness). If you are reading this and are interested, would you check for any errors? Thank you for reading!
-Proof-
Take $z\geq2\in\mathbb{Z}$ where z is not prime.
Then $$ \exists\;a_{01},a_{02}\geq2\;s.t.\;a_{01}a_{02}=z$$
If both factors are prime, than we are done.
If not, then $$\exists\;a_{11},a_{12}\geq2\;s.t. a_{11}a_{12}=a_{0i_{0}}$$ where $i_{0}=1$ or $2$
If $a_{11},a_{12}$ prime, we are done.
If one is not, then $$ \exists a_{21}a_{22}\geq2\; s.t.\; a_{21}a_{22}=a_{1i_{1}}\; where\; i_{1}=1 \;or\; 2 $$
The process continues ad infinitum $\iff \nexists $ a factorization into primes for z
Suppose this process would continue ad infinitum.
Where $ j_{n}=1 \;or\; 2 \;and\; j_{n}\neq i_{n} $
$$z>a_{0j_{0}}\cdot a_{1j_{1}}\cdot a_{2j_{2}}\cdot....=\prod_{n\geq0}^{\infty}a_{nj_{n}} $$
But $$a_{nj_{n}}\geq2\; \forall\;n$$
Thus $$ z>\infty \;and\; z=\infty $$
But we took z finite thus this process must stop at some point, and thus $\exists$ prime factorization for all positive integers
