Single Pole Amplifier
This is nearly a mathematical tautology, like saying that a positive number cannot go below zero.  A one pole filter can introduce a phase shift which is at most ninety degrees. 

Instability requires positive feedback at some multiple of 360 degrees with at least unity gain.  But amplifiers use negative feedback. A two-pole amplifier is under threat, because it poles can potentially create up to 180 degrees of phase shift, which will combine with the 180 degree of phase inversion from negative feedback to create in-phase positive feedback.

In fact, instability under negative feedback begins at less than 180 degree of phase shift. As a rule of thumb, a safety margin (called phase margin) of at least 45 degrees is required, so 135 degrees of shift or less. A commonly used stability criterion is that at any frequency where there is unity gain or greater, the phase shift should be 135 degrees or less. If there is any frequency where this is not true, the amplifier may either self-oscillate, or at least exhibit overshoot and ringing: damped oscillations which are stimulated by input, and take several swings to die down.

One way in which amplifiers with multiple poles are stabilized is with the help of capacitors which create a "dominant pole" whose frequency roll-off is so great that the poles at higher frequencies basically do not matter (the gain is squashed at those frequencies). The amplifier basically "looks" like a single pole one.