  Compressive sensing predicts that sufficiently sparse vectors can be
recovered from highly incomplete information. Efficient recovery methods such
as $\ell_1$-minimization find the sparsest solution to certain systems of
equations. Random matrices have become a popular choice for the measurement
matrix. Indeed, near-optimal uniform recovery results have been shown for such
matrices. In this note we focus on nonuniform recovery using Gaussian random
matrices and $\ell_1$-minimization. We provide a condition on the number of
samples in terms of the sparsity and the signal length which guarantees that a
fixed sparse signal can be recovered with a random draw of the matrix using
$\ell_1$-minimization. The constant 2 in the condition is optimal, and the
proof is rather short compared to a similar result due to Donoho and Tanner.
