The image you posted in the sister comment made me see what's going on. You are arguing according to the Aristotelian theory of projectile motion, while the other commenters assume the Galilean theory. See e.g. this link: http://www.met.reading.ac.uk/pplato2/h-flap/phys2_2.html#sec... .

(In the presence of strong air resistance, the Aristotelian theory is not a bad approximation. If there is air resistance, a falling object will indeed gradually slow down to a terminal velocity, like you write in your last paragraph.)