Proper time differences

  From various sources I gather,

dt = "gamma" d"tau".

  Where t is the coordinate time in the rest frame, "gamma"
  is the Lorentz gamma factor and "tau" is the proper time.

  Now, if "gamma" is constant, I think we can replace the "d"
  by "D" (triangle which is flat at its bottom), i.e., we can
  use finite difference instead of infinitesimal ones:

Dt = "gamma" D"tau".

  I believe 0<="gamma"<=1, so, for an example, we can assume
  "gamma" to be 0.5:

Dt = 0.5 D"tau",

  which means just,

D"tau" = 2 Dt.

  So, that would mean: For a moving thing the proper time
  difference D"tau" (I assume: between two fixed events) is
  /larger/ than the coordinate time difference.

  But since falling muons live longer, the proper time distance
  should be /smaller/, not larger!

  What's wrong here? TIA!