It’s a terribly organized and presented proof, but I think it’s basically right (although it’s skipping over some algebraic details, which is common in proofs). To spell it out:
Fix any x and y. We then have,
x2−2xy+y2=(x−y)2≥0.
Adding 2xy to both sides,
x2+y2≥2xy.
Therefore, if (by assumption in that line of the proof) g(x)>x2 and g(y)≥y2, we’d have,
g(x)+g(y)>x2+y2≥2xy,
which contradicts our assumption that g(x)+g(y)≤2xy.
Thanks. When it’s written as g(x)+g(y)>x2+y2≥2xy, I can see what’s going on. (That one intermediate step makes all the difference!)
I was wrong then to call the proof “incorrect”. I think it’s fair to call it “incomplete”, though. After all, it could have just said “the whole proof is an exercise for the reader”, which is in some sense correct I guess, but not very helpful (and doesn’t tell you much about the model’s ability), and this is a bit like that on a smaller scale.
(Although, reading again, ”...which contradicts the existence of y∗ given x” is a quite strange thing to say as well. I’m not sure I can exactly say it’s wrong, though. Really, that whole section makes my head hurt.)
If a human wrote this, I would be wondering if they actually understand the reasoning or are just skipping over a step they don’t know how to do. The reason I say that is that g(x)+g(y∗)>2xy∗ is the obvious contradiction to look for, so the section reads a bit like “I’d really like g(y∗)<(y∗)2 to be true, and surely there’s a contradiction somehow if it isn’t, but I don’t really know why, but this is probably the contradiction I’d get if I figured it out”. The typo-esque use of y instead of y∗ bolsters this impression.
It’s a terribly organized and presented proof, but I think it’s basically right (although it’s skipping over some algebraic details, which is common in proofs). To spell it out:
Fix any x and y. We then have,
x2−2xy+y2=(x−y)2≥0.
Adding 2xy to both sides,
x2+y2≥2xy.
Therefore, if (by assumption in that line of the proof) g(x)>x2 and g(y)≥y2, we’d have,
g(x)+g(y)>x2+y2≥2xy,
which contradicts our assumption that g(x)+g(y)≤2xy.
Thanks. When it’s written as g(x)+g(y)>x2+y2≥2xy, I can see what’s going on. (That one intermediate step makes all the difference!)
I was wrong then to call the proof “incorrect”. I think it’s fair to call it “incomplete”, though. After all, it could have just said “the whole proof is an exercise for the reader”, which is in some sense correct I guess, but not very helpful (and doesn’t tell you much about the model’s ability), and this is a bit like that on a smaller scale.
(Although, reading again, ”...which contradicts the existence of y∗ given x” is a quite strange thing to say as well. I’m not sure I can exactly say it’s wrong, though. Really, that whole section makes my head hurt.)
If a human wrote this, I would be wondering if they actually understand the reasoning or are just skipping over a step they don’t know how to do. The reason I say that is that g(x)+g(y∗)>2xy∗ is the obvious contradiction to look for, so the section reads a bit like “I’d really like g(y∗)<(y∗)2 to be true, and surely there’s a contradiction somehow if it isn’t, but I don’t really know why, but this is probably the contradiction I’d get if I figured it out”. The typo-esque use of y instead of y∗ bolsters this impression.