Hasse's bound

Consider the Frobenius endomorphism $\varphi: E(\overline{\mathbb{F}_p}) \to E(\overline{\mathbb{F}_p})$, $\varphi(x,y) = (x^p, y^p)$. Its fixed points are exactly $E(\mathbb{F}_p)$, and one shows $\#E(\mathbb{F}_p) = \deg(\varphi - 1) = p+1-t$ where $t = \varphi + \hat\varphi$ is the trace of Frobenius acting on the endomorphism ring. The degree map on endomorphisms of $E$ is a positive-definite integer-valued quadratic form (it satisfies $\deg(m\varphi+n) \ge 0$ for all integers $m,n$, with equality only when $m\varphi+n=0$), and $\deg\varphi = p$. Expanding $\deg(m\varphi+n) = m^2 p + mnt + n^2 \ge 0$ as a quadratic form in $m,n$ forces its discriminant to be non-positive: $t^2 - 4p \le 0$, i.e. $|t|\le 2\sqrt p$. Since $\#E(\mathbb{F}_p) - (p+1) = -t$, this is exactly the claimed bound.