Showing a set is closed, question from real analysis

Question

Let $\mathbb{X}=\{1,2,3\}$ and let $\mathbb{P}$ be the set of all probabilities on $\mathbb{X}$. $\,\,$ Let, $V:\mathbb{P}\to \mathbb{R}$ be defined as $V(p)=(1+p_1)^2+(2+p_2)^2+(3+p_3)^2$. Show that $\forall\,P,Q,R\in\mathbb{P}$, $\{\lambda\in[0,1]:V(\lambda P+(1-\lambda) Q)\geq V(R)\}$ is closed. $$\\$$I am wondering if there is a easier way to approach this other than just substituting the values in $V(\lambda P+(1-\lambda) Q)\geq V(R)$ and solving there on. To be honest I still haven't tried the mechanical way of substitution and solving, but, I can give it a try if so suggested.

Answer 1

Fix some $P,Q,R\in\mathbb{P}$, and define $f:[0,1]\to\mathbb{R}$ by $\lambda\mapsto V(\lambda P+(1-\lambda)Q)-V(R)$. Directly from the definition, $f$ is continuous, and the subset in question is just $f^{-1}(\mathbb{R}_{\ge0})$. It is closed, as the inverse image of a closed subset under a continuous function.