Closedness of the closed half-space Suppose we have a hyperplane $H(p, \alpha) = \{x \in \mathbb{R}^n
\mid p \cdot x = \alpha\}$ , then how do we prove that one of the corresponding closed half-spaces, $H^*(p, \alpha) = \{x \in \mathbb{R}^n \mid p \cdot x \leq α\}$ is indeed closed? 
For every $x$ that is an element of  the complement of  $H^*(p, \alpha)$, can we find an associated $r > 0$ such that a ball centered at $x$ with the radius $r$ is a subset of the complement of $H^*(p, \alpha)$ and therefore proving that the complement of $H^*(p, \alpha)$ is open and $H^*(p, \alpha)$ is indeed closed. I am having trouble finding this $r > 0$. 
Thank you for your help.

Let $p$ and $a$ be given, and suppose that $p\cdot x > a$ for some $x$. The goal is to show that there exists $\delta > 0$ such that $p \cdot (x+y) > a$ for all $y$ satisfying $|y| < \delta$. If $\epsilon = p\cdot x -a$, then everything works if $|p\cdot y| < \epsilon/2$, which is definitely accomplished if $|p||y| < \epsilon/2$. So, $\delta=\epsilon/2(|p|+1)$ works.
