There seem to be a surprising number of posters here who think that a head wind allows one to climb higher than in still air.
Guys,
If you have a speed of 275mph in still air your true airspeed and ground speed will both be 275.
If you are in a 25mph headwind your true air speed will be 275mph and your ground speed will be 250mph. Your speed through the air and the speed of the air over your wings and the lift provided by your wings is not changed by a head wind.

If you have a speed of 275mph in still air your true airspeed and ground speed will both be 275. Not always, depends entirely on the existing density altitude! The higher the altitude, the ground speed in "zero" wind, the ground speed will always be greater than the true airspeed.
Aircraft flight instruments, however, don't compute true airspeed as a function of groundspeed and windspeed. They use impact and static pressures as well as a temperature input. True airspeed is equivalent airspeed that is corrected for pressure altitude and temperature (which define density). The less dense air at altitude allows the aircraft to move faster over the ground but because of the less dense air, the IAS would be about the same, and the true airspeed, after correcting for pressure altitude and temperature, would be greater and in zero wind, the ground speed would always be greater.