  We prove the following conjecture, due to Tomaszewski (1986): Let $X=
\sum_{i=1}^{n} a_{i} x_{i}$, where $\sum_i a_i^2=1$ and each $x_i$ is a
uniformly random sign. Then $\Pr[|X|\leq 1] \geq 1/2$. Our main novel tools are
local concentration inequalities and an improved Berry-Esseen inequality for
Rademacher sums.
