Although the sequence of greedy approximants associated with the Schauder expansion of a function, $f$, continuous on $[0,1]$, may fail to converge, there always will be a continuous function, arbitrarily close to $f$, whose Schauder expansion does have a convergent sequence of greedy approximants. Further examination of this problem shows that the same sort of proposition is valid for a multitude of subsystems of the Schauder system.
"Greedy Approximation in Certain Subsystems of the Schauder System." Real Anal. Exchange 34 (1) 227 - 238, 2008/2009.