  We prove that in every separable Banach space $X$ with a Schauder basis and a
$C^k$-smooth norm it is possible to approximate, uniformly on bounded sets,
every equivalent norm with a $C^k$-smooth one in a way that the approximation
is improving as fast as we wish on the elements depending only on the tail of
the Schauder basis.
  Our result solves a problem from the recent monograph of Guirao, Montesinos
and Zizler.
