Saddle Point Property

I felt dumb while trying to derive this, so here it is. Sourced from page 129 of Braess FEM textbook.

Let $X, M$ be two Hilbert spaces, and $a: X \times X \to \mathbb{R}, \, b: X \times M \to \mathbb{R}$ continuous bilinear forms.
Assume that $a$ is symmetric and that $a(u, u) \ge 0$ for all $u \in X$.
Let $f \in X’, g \in M’$ and let
$$\begin{align*} \mathcal{L}(v, \mu) &= \frac{1}{2}a(v, v) – \langle f, v \rangle + (b(v, \mu) – \langle g, \mu\rangle) \end{align*}$$
which is simply the Lagrangian of a constrained minimization problem.
Assume that $(u, \lambda)$ satisfies
$$\begin{align} a(u, v) + b(v, \lambda) &= \langle f, v \rangle \qquad &\forall v \in X, \\ b(u, \mu) &= \langle g, \mu \rangle \qquad &\forall \mu \in M, \end{align}$$
then one has the saddle point property
$$\begin{align*} \mathcal{L}(u, \mu) \le \mathcal{L}(u, \lambda) \le \mathcal{L}(v, \lambda) \qquad \forall (v, \mu) \in X \times M. \end{align*}$$

The first inequality is actually an equality by noting that $\mathcal{L}(u, \mu) = \frac{1}{2}a(u, u) – \langle f, u \rangle= \mathcal{L}(u, \lambda)$ by using (2).
For the other inequality, let $v = u + w$, then
$$\begin{align*} \mathcal{L}(v, \lambda) = \mathcal{L}(u + w, \lambda) &= \frac{1}{2}a(u + w, u + w) – \langle f, u + w \rangle + (b(u + w, \lambda) – \langle g, \lambda\rangle) \\ &= \mathcal{L}(u, \lambda) + \frac{1}{2}a(w, w) – \langle f, w \rangle + a(u, w) + b(w, \lambda) \\ &= \mathcal{L}(u, \lambda) + \frac{1}{2}a(w, w) \ge\mathcal{L}(u, \lambda) \end{align*}$$
where (1) and (2) are used.