Proof II does gloss over a few details though; it should note that that prime factorization is unique for each integer.
Another clever proof is the geometric proof of the AM-GM inequality:
Draw a semicircle, and choose an arbitrary point on the diameter. Draw the inscribed right triangle with the diameter as the hypotenuse. Now draw the altitude perpendicular to the hypotenuse, bisecting the hypotenuse into to segments, A and B. Note that the radius is the arithmetic mean of A and B, and the altitude is the gemometric mean of A and B.
Proof II does gloss over a few details though; it should note that that prime factorization is unique for each integer.
Another clever proof is the geometric proof of the AM-GM inequality:
Draw a semicircle, and choose an arbitrary point on the diameter. Draw the inscribed right triangle with the diameter as the hypotenuse. Now draw the altitude perpendicular to the hypotenuse, bisecting the hypotenuse into to segments, A and B. Note that the radius is the arithmetic mean of A and B, and the altitude is the gemometric mean of A and B.
Clearly, radius >= altitude.