Delta-convergent sequences that vanish at the support of the limit Dirac delta function

The known limit representations of the Dirac delta function involve delta sequences that do not vanish at the support of the limit Dirac delta function. However, Galapon (2009 Proc. R. Soc. A 465 71) assumes that a delta sequence may vanish at the support of the limit Dirac delta function for all finite values of the limit parameter. Here, such delta sequences are shown to exist by construction.