Real Analysis: proving the triangle inequalities

I am trying to prove one of the triangle inequalities.

[] are going to be my absolute value symbols, sorry, i don't have mathematica on this computer with the symbols.

I am proving that [[x]-[y]]<=[x-y].

i tried this: we know -[x]<=x<=[x] and -[y]<=y<=[y]

if i subtract these i get: -[x]--[y]<=x-y<=[x]-[y]

then you simplify and get -([x]-[y])<=x-y<=([x]-[y])

so, [x-y]<=[[x]-[y]].

I thought i was on the right track but then i realized that the inequality was pointing in the wrong direction. Did i make some computational error or am i going about this the wrong way? I appreciate any hints on where I might start. thanks!