Quadratic Equation positivity

Question

$Lemma-4.4.2:$ If $a,b,c$ are real numbers such that $a>0$ and $a\lambda^2+2b\lambda+c\geq0$ for all real number $\lambda$, then $b^2\leq ac$.

This is from "Topics in Algebra by I.N Herstein"

$Question:$ Here in this lemma, if we don't assume $a>0$ then it is still valid. Because $a\lambda^2+2b\lambda+c\geq0$ for all $\lambda$, implies $a>0$.So assuming this extra condition is immaterial. Please correct me.

Answer 1

We could also have $a=b=0$, but the conclusion still holds in that case. So the condition is superfluous, but it isn't quite empty because it could possibly not be the case. You could also just as well specify that $a\neq 0$ or $b\neq 0$.

Answer 2

On the contrary, the condition $a>0$ is necessary. Otherwise, we would not have the quadratic function to be $\ge 0$ identically.