  We prove the existence of exponentially localised and time-periodic solutions
in general nonlinear Hamiltonian lattice systems. Like normal modes, these
localised solutions are characterised by collective oscillations at the lattice
sites with a uniform time-dependence. The proof of existence uses the
comparison principle for differential equations to demonstrate that at each
lattice site every half of the fundamental period of oscillations, contributing
to localised solutions of the nonlinear lattice, is sandwiched between two
oscillatory states of auxiliary linear equations. For soft (hard) on-site
potentials the allowed frequencies of the in-phase (out-of-phase) localised
periodic solutions lie below (above) the lower (upper) value of the linear
spectrum of phonon frequencies. By varying a control parameter the exponential
decay of the localised states can be tuned continuously.
