Show that the upper envelope of a bounded function is upper semi continuous directly

Show that the upper envelope of a bounded function is upper semi
continuous directly

Definition 1:
A real valued function $f$ is said to be upper semicontinuous at a point
$p$ if: $$f(p) \geq \limsup_{x \rightarrow p} f(x) $$
Definition 2:
Let $f$ be a bounded real valued function on $[a,b]$. Define the upper
envelope $h$ of $f$ as:
$$ h(y) = \inf_{\delta >0} \sup_{|x-y|<\delta} f(x)$$
The question: If $f$ is a real valued bounded function on $[a,b]$, show
that the upper envelope of $f$ is upper semicontinuous.
Background:
This is a sample qualifying exam question. In the past, this question is
sometimes asked with a part a which says that a function is upper
semicontinuous iff the sets $\{x: f(x) < \lambda\}$ are open for each
$\lambda \in \mathbb{R}$. This characterization is not too difficult to
prove. Also, the desired result is not too difficult to deduce from this
characterization. For example, see:
Upper semi-continuity and lower semi-continuity of particular functions
However, this approach is unsatisfactory. In particular, this problem has
shown up sans part a before on the qualifying exam. I'm looking for a
proof that takes a bounded real valued function $f$, and from Definition 2
deduces Definition 1 directly. Every attempt I make, I get lost in chasing
infs and sups through inequalities. Any help would be much appreciated.