My proof that $T(n) = 2T(n/2) + n$ is $\Omega(n \lg n)$:
Assume $T(m) \geq cm \lg m$ for $m < n$.
$T(n) \geq cn \lg(n/2) + n = cn \lg n - cn + n \geq cn \lg n$, for $0 < c \leq 1$.
Since we just have to find there exists a positive constant c, are we done?