How does $\inf_{c \in \mathbb{R}} \lVert u - c \rVert_{L^2} \le \lVert \nabla u \rVert_{L^2}$ imply this inequality? Let $M$ be a compact Riemann manifold with boundary. I want to know, given the inequalities $$
  \vert T u \vert_{H^{1/2} (\partial M)} \le C \lVert \nabla u \rVert_{L^2(M)} + \lVert u \rVert_{L^2(M)},
$$
and 
$$
  \inf_{c \in \mathbb{R}} \lVert u - c \rVert_{L^2(M)}
\le \lVert \nabla u \rVert_{L^2(M)}.
$$
how do I obtain 
$$|Tu|_{H^{\frac 12}(\partial M)} \leq C\lVert \nabla u \rVert_{L^2(M)}?$$
(The $C$ here can be a different constant) 
This is apparently true from an answer on Mathoverflow.. I asked the author but he hasn't responded. Does anyone know how to get it??!

Since constant functions have zero $H^{1/2}$ seminorm, it follows that 
$$\vert T u \vert_{H^{1/2} (\partial M)} =  \vert (T u) -c \vert_{H^{1/2} (\partial M)} = \vert T (u -c) \vert_{H^{1/2} (\partial M)}\tag{1}$$
for every $c\in\mathbb{R}$. (Trace operator commutes with adding a constant, because the trace of a constant function is that constant function.)
Therefore,  the first inequality you cited yields
$$\vert T u \vert_{H^{1/2} (\partial M)} \le C \lVert \nabla u \rVert_{L^2(M)} + \lVert u-c \rVert_{L^2(M)}\tag{2}$$
Take infimum over $c$ and use the Poincaré inequality (which is missing a constant in your question): 
$$\vert T u \vert_{H^{1/2} (\partial M)} \le C \lVert \nabla u \rVert_{L^2(M)} + C'\lVert \nabla u \rVert_{L^2(M)}\tag{3}$$
