  In this paper, a direct rigorous mathematical proof of the Gregory-Laflamme
instability for the five-dimensional Schwarzschild black string is presented.
Under a choice of ansatz for the perturbation and a gauge choice, the
linearised vacuum Einstein equation reduces to an ODE problem for a single
function. In this work, a suitable rescaling and change of variables is applied
which casts the ODE into a Schr\"odinger eigenvalue equation to which an energy
functional is assigned. It is then shown by direct variational methods that the
lowest eigenfunction gives rise to an exponentially growing mode solution which
has admissible behavior at the future event horizon and spacelike infinity.
After the addition of a pure gauge solution, this gives rise to a regular
exponentially growing mode solution of the linearised vacuum Einstein equation
in harmonic/transverse-traceless gauge.
