I have written a formal proof for the first time and am looking for input on any errors I may have made or general tips thanks.
Theorem: For all real numbers x and y, $$x^2 + y^2 + 1 \geq xy+x+y.$$
Proof: We prove that the inequality $$x^2 + y^2 + 1 \geq xy+x+y.$$(1)
holds for all $x,y\in \mathbb{R}$ . By algebra, this inequality can be written in the equivalent form: $$\frac{x^2+y^2}2+\frac{x^{2}+1}2+\frac{y^2+1}2 \geq xy + x+y$$
(2)
Recall, the inequality of arithmetic and geometric means in its general case is applicable to all nonnegative real numbers and is:
$$\frac{x_1+x_2+...+x_n}{n} \geq \sqrt[n]{x_{1}x_{2}...x_{n}}$$
We can apply this to the left hand size of (2) since x and y are squared and any square is nonnegative.
If we do this the AM-GM of the left hand side of (2) is equivalent to:
$$\frac{x^2+y^2}2+\frac{x^{2}+1}2+\frac{y^2+1}2 \geq \sqrt[2]{x^2y^2}+\sqrt[2]{x^2(1)}+\sqrt[2]{y^2(1)}$$ (3)
When simplified the right-hand side of the inequality (3) is equivalent to the right hand side of (1): $$\frac{x^2+y^2}2+\frac{x^{2}+1}2+\frac{y^2+1}2 \geq xy + x+y$$
Q.E.D.