How do I make this simple proof better (and more correct?) Let $x$ and $y$ be real numbers. If $x\cdot{y}>\frac{1}{2}$, then $x^2+y^2>1$.
Proof: We will prove with the direct method. Let $x$ and $y$ be real numbers. Since
$$ x\cdot{y}>\frac{1}{2} $$
it follows that
$$ 2xy>1,$$
which means
$$x^2+y^2 \geq 2xy.$$
Therefore,
$$ x^2+y^2>1. $$
 A: Since $x,y\not=0\Rightarrow x^2,y^2\gt 0$, by AM-GM inequality, we have
$$x^2+y^2\ge 2\sqrt{x^2y^2}=2|xy|=2xy\gt 2\cdot \frac 12=1.$$
A: Write $0\le (x-y)^2=x^2-2xy+y^2$, hence $1<2xy\le x^2+y^2$.
A: I'm probably going to be a little grammatically picky, but I personally think proper grammar is good in a proof, (and an analysis lecturer drilled it into our class to use 'proper English sentences' where the maths would read as part of the sentence).


*

*I would remove the comma after the first line of mathematical notation say something like 'we have/get', instead of 'therefore'.

*'If we substitute...' (no and)
A: The only problem is the "which means". 
$2xy>1$ doesn't imply $x^2+y^2≥2xy$ in any way. 
$x^2+y^2≥2xy$ is true because $x^2-2xy+y^2 = (x-y)^2 ≥ 0$, which means $x^2+y^2≥2xy$. 
A: There is a much simpler proof.
Since $xy > \frac{1}{2}$, $x$ and $y$ must be non zero numbers, which are both positive or negative.
Suppose $x>0$ and $y>0$.
If $x>1$, then also $x^2>1$, and $x^2 +y^2>x^2>1$ is obvious.
Take $0<x\leq 1$ and $y>0$.
Since $0<x\leq 1$, the function $f$ with $f(x)=\sqrt{1-x^2}$   is well defined.  Let  $g$ be the function with $g(x)=\frac{1}{2x}, \ 0<x\leq 1$.
It can be easily proved that the function h with $h(x)=-x+\sqrt{2}, \ 0<x\leq 1$ 
      has a graph which is a line tangent to the graphs of $f$ and $g$ at the point
      $(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$.
One can also verify easily that $f$ is a concave function and that $g$ is a convex funtion. Therefore the tangent line $h$ is above the graph of $f$ and below the graph of $g$. 
      So if  $0<x\leq 1$,the previous remark implies that  $f(x)\leq h(x)\leq g(x)$ and  $\sqrt{1-x^2}\leq \frac{1}{2x}$.
Now, the proposition we want to prove is obvious: If $xy > \frac{1}{2}$, then $y>\frac{1}{2x}$, 
      and $\frac{1}{2x} \geq \sqrt{1-x^2}$ from the previous inequality, hence 
      $y> \frac{1}{2x} \geq \sqrt{1-x^2}$. Consequently, $y>\sqrt{1-x^2}$, which implies
      $y^2>1-x^2$ and therefore $x^2+y^2>1.$
The case $x<0$ and $y<0$ is similar.
