Why are $C^0(\mathbb{R})$ and $C^{0,0} (\mathbb{R})$ the same spaces?

Question

$C^0(\mathbb{R})$ has the norm $\Vert f \Vert_{C^0(\mathbb{R})}$.

$C^{0,0} (\mathbb{R})$ has the norm $\Vert f \Vert_{C^0(\mathbb{R})} + \sup_{x,y \in \mathbb{R}, x \neq y} |f(x) - f(y)|$.

I don't see how these are the same norms?

Answer 1

Hint: $$ |f(x)-f(y) | \leq |f(x)| + |f(y)| \leq 2 \sup |f|. $$ Therefore the norms are equivalent.