Let $U\in C^1(\mathbb R\times\mathbb R^n,\mathbb R)$.
During class, the lecturer stated that it is possible to find two functions $a_1, a_2:\mathbb R\to\mathbb R$ such that $$ U(x, y/|y|)\ge \inf_{|z|=1} U(x, z)\ge a_1(x)$$ and $$ U(x, y/|y|)\le \sup_{|z|=1} U(x, z)\le a_2(x).$$
Anyone could please help me to understand why is that true? I do not even understand why he wrote that $$\inf_{|z|=1} U(x, z)\le U(x, y/|y|)\le \sup_{|z|=1} U(x, z).$$
Thank you in advance.