Supremum of quotient of two functions This should be an easy question, but I am kind of unsure today.
I have a compact subset $I \subset \mathbb{R}$ and two real valued functions $f:\mathbb{R} \to \mathbb{R}$ and $g:\mathbb{R} \to \mathbb{R}$, where $g(x) >0$.
Do I then have:
$$\frac{\sup_{t\in I}|f(t)|}{\sup_{t\in I}g(t)} \leq \sup_{t\in I} \frac{|f(t)|}{g(t)}?$$
I claim yes, since $\sup_{t\in I}g(t) \geq g(t)$  for all $t \in I$ from which the assertion then already should follow. 

Hint: If $u$ and $v$ are nonnegative on $I$, then $\sup\limits_I (uv) \leq (\sup\limits_I u)( \sup\limits_I v)$.
Consider $u=g$ and $v=|f|/g$.
