Removing artifact in Plot[1-(1-x)^Floor[2/x],{x,0,1/20}]

The input

Plot[1-(1-x)^Floor[2/x],{x,0,1/20}]

yields

where the white space we see below the curve near $x=0.01$ is an artifact. How can we fix this simply? WorkingPrecision -> 999 does not help, the issue is the way $x$ is sampled.

Plot[1-(1-x)^Floor[2/x],{x,0,1/20}, PlotPoints -> 10^4]