The authors consider the second order Hamiltonian system \(\ddot{u}+V_u(t,u)=0\) where \(V:\mathbb R\times\mathbb R^N\rightarrow\mathbb R\) is \({\mathcal{C}}^2\), \(T\)-periodic in \(t\), and \(V_u\) is globally bounded. Further conditions on \(V\) for \(|u|\rightarrow\infty\) are not required. They prove several results on the existence of \(T\)-periodic solutions under additional conditions on \(V_u(t,u)\) for \(u\) in annuli \(\{u:c\leq|u|\leq d\}\). The proofs use the Conley index applied to the negative gradient flow of the action integral, restricted to finite-dimensional subspaces \(E_n\subset W^{1,2}(\mathbb R/T\mathbb Z,\mathbb R^N)\) which come from an approximation scheme. The conditions on \(V_u\) yield isolating blocks \((B_n, B^-_n)\) for the induced flow on \(E_n\) which are bounded uniformly in \(n\). The nontriviality of the Conley index \([B_n/B^-_n]\) together with a standard passage to the limit as \(n\rightarrow\infty\) gives the solution.