Spaces of sequences not converging to zero

Let \(E\) be a Banach space (or a Banach lattice), let \(\tau\) be a vector topology on \(E\) and let \({\bf x}\) be a sequence (or a positive sequence) in \(E\) not converging to zero with respect to \(\tau\). We show how to construct infinite dimensional Banach spaces (or Banach lattices) consisting, up to the origin, of sequences in \(E\) not converging to zero with respect to \(\tau\) and containing a subsequence of \({\bf x}\). Plenty of applications to Banach space theory and to Banach lattice theory are provided.