Why is thrust said to be constant over speed for a jet engine?
There are several effects which in combination make constant thrust a good approximation at subsonic speed.
Thrust is created by accelerating a working mass in opposite direction. Net thrust is the difference between the impulse of the air flowing towards the engine and the combined impulse of burnt fuel and the air exiting the engine (and propeller, if one is fitted), derived after the time. Since that impulse is the product of mass and speed, you can either accelerate a large mass by a small speed difference, like a propeller does, or a small mass by a large speed difference, like a turbojet does.
When flying faster, the entry impulse of a propeller quickly grows large relative to the exit impulse, so thrust goes down with the inverse of speed. On the other hand, the high exit speed of a turbojet results only in a small increase of the entry impulse relative to the exit impulse while speed increases.
But if that were all, even the thrust of a turbojet engine would drop when speed increases. But there is a second effect which helps to let thrust grow with speed. With the square of speed, to be precise. That is the ram effect which helps to precompress the air entering the engine. At subsonic speed, this just about compensates for the loss of thrust: At low speed, the growing entry impulse lets thrust drop a bit but at higher subsonic speed the ram effect becomes larger and raises thrust again, such that a constant thrust becomes a good approximation. However, at supersonic speed the ram effect becomes dominant and thrust grows with speed squared – until the absolute internal pressure becomes too high so the engine must be throttled (or the aircraft needs to fly higher) or the shock losses in the intake become too large and thrust drops again.