Ideal Gases under Adiabatic Compression

I'm having trouble understanding what happens to the internal energy of an ideal gas being compressed adiabatically.

If DU = DQ + DW,

then as we do work PdV compressing the gas, since in adiabatic processes DQ=0, W the change in internal energy is non-zero, so U must increase.

But if we're talking about an ideal gas, as I keep hearing, (such as in this lecture where we're told an ideal gas shouldnt increase in temperature when we compress it in a bicycle pump:)

http://www.youtube.com/watch?v=g14939TMTCE#t=36m30s

U is a function only of T, and so T ought not to vary as we increase the pressure, because it ought to be compensated for by a (countering) decrease in V, in accordance with the Ideal Gas Law: PV = nRT.

So should U stay constant or should it increase under adiabatic compression if the gas is ideal?

Can someone help me with this?

Thanks a lot!