Duality Proof - Consumer Theory For solving this question, which assumptions/axioms should be used and should Lagrange multipliers be equal or different? 

Think of an agent with utility function u() and wealth w. Assume that u() is continuus and represents a locally non-satiated preference relation. The utility maximization problem of this agent can be stated as:

max  u(x)
 s.t. p.x ≤ w

On the other hand, when the agent’s constraint is to attain a utility level h, her expenditure minimization problem can be stated as:

min  p.x 
s.t. u(x) ≥ h

Show that if x* is optimal in the utility maximization problem when w > 0, then x* is optimal in the expenditure minimization problem when the constraint is to attain a utility level u(x∗).
Let us prove it by contradiction. Suppose that $x^{*}$ is optimal for the first problem but not for the second. Since $x^{*}$ is not optimal for the expenditure minimization problem, there exists a bundle $x$ such that 
\begin{equation*}
u(x) \geq u(x^{*}) \text{ and } p \cdot x &lt; p \cdot x^{*}
\end{equation*}

Since the function $y \rightarrow p.y$ is continuous, and $p \cdot x &lt; p \cdot x^{*}$, there exists $\epsilon&gt;0$ such that $\|y-x\|&lt;\epsilon \Rightarrow p \cdot y &lt; p \cdot x^{*}$. In addition, since preferences are locally nonsatiated, there exists $y$ such that $\|y-x\|&lt;\epsilon$ and $u(y)&gt;u(x)$. Take such a $y$. We have \begin{equation*}
u(y) &gt; u(x) \geq u(x^{*}) \text{ and } p \cdot y &lt; p \cdot x^{*} \leq w
\end{equation*}

Or, in other words,
\begin{equation*}
u(y)&gt;u(x^{*}) \text{ and } p.y &lt; w
\end{equation*}
which contradicts the assumption that $x^{*}$ is a solution of the utility maximization problem.