I have been having trouble with how to go forward with a proof for about three days now. I know the basic structure of the proof, but can't seem to construct it.
Basically, I am trying to do a proof by contradiction for the following:
Say $u: x \rightarrow \mathbb{R}$ has no local maxima. Let $p \in \mathbb{R^l}_{++}$ and $w>0$. Show that if $x^*$ is a solution to the maximization problem:
$$\max_x \ (u(x)) \ \text{s.t.} \ x \in B(p) = [x \in \mathbb{R^l}_{++} : p \cdot x \leq w]$$ then for all $y \in \mathbb{R^l_{++}}$ such that $u(y) \geq u(x^*)$, then it must be that $p \cdot y \geq w$
So I'm supposed to do this proof by contradiction, (suppose we have $y \in \mathbb{R^l_{++}}$ such that $u(y) \geq u(x^*)$, then it must be that $p \cdot y < w$) and use the fact that $u$ doesn't have a local max implies that the function $u$ is locally non-satiated:
$$\forall y \in \mathbb{R^l_{+}, \forall \epsilon > 0, \exists {y'} \in \mathbb{R^l_{+}}} \ \text{s.t.} \ \|y - y'\| < \epsilon \ \text{and} \ y' \succ y$$
But I've been stuck for a while now. Any help would be appreciated.