Let $f:[0,\infty)\rightarrow\mathbb{R}$ be a right-continuous function. Consider the set $$ E=\left\{x\geq 0\left|\sup_{0\leq y\leq x}f(y)\geq K\right.\right\}, $$ where $K$ is a constant. It is pretty intuitive that the right-continuity of $f$ makes $E$ a closed set, that is $x_{E}=\inf E\in E$. Any suggestion? Below I report my attempt.
After some thoughts, I came up with this proof. Hope that someone can verify its correctness.
