Why is the lagrange dual function concave?

In a book I'm reading (Convex Optimization by Boyd and Vandenberghe) it says

I'm struggling to understand the last sentence. Why can one conclude concavity from having a pointwise infimum of a family of affine functions?

Because the Lagrangian $L(x,\lambda,\mu)$ is affine in $\lambda$ and $\mu$, the Lagrange dual function $d(\lambda,\nu) = \inf_{x\in \mathcal{D}}L(x,\lambda,\nu)$ is always concave because it is the pointwise infimum of a set of affine functions, which is always concave. (You can also show that the supremum of a set of convex functions is convex.)

The book referenced is Convex Optimization by Boyd and Vandenberghe. To better see the "pointwise infimum", consider a slight change/abuse of notation: $L_x(\xi) = L(x, \lambda, \nu)$ where $\xi = (\lambda, \nu)$. For a fixed $x$, $L_x(\xi)$ is affine in $\xi$ so $\{L_x(\xi) \,:\, x \in \mathcal{D}\}$ is a family of affine functions and its pointwise infimum is $$g(\xi) = \inf_x \,\{L_x(\xi)\,:\,x\in\mathcal{D}\}$$ Now we can use @A.Γ.'s pointer to show that $g$ is concave by showing the epigraph of $-g$ is convex. For a given $\xi$, we have $g(\xi) \le L_x(\xi)$ for any $L_x$ from the family so $(\xi, -g(\xi))$ is always "above" $(\xi, -L_x(\xi)$) hence $\rm{epi}(-g) \subset \bigcap_x \rm{epi}(-L_x)$.