Poles and Stability of System

Why do we say that poles should be within unit circle for the system to be stable ?

let the Transfer function which is the designed filter in our system have a impulse response h(n), the system is stable only if h(n) tends to zero as n tends to infinity.

In terms of Z transform, consider a causal impulse response of the form 

                                               ,n=0,1,2...

If r>1 , then the amplitude envelope will increase exponentially as ,

the Signal  has the Z-Transform,

                  

                   

                    taking in ,  or 

thus the Transfer function consists of a single pole at  when  , now consider   , the poles of H(z) moves out of unit circle and the impulse response has exponential increasing amplitude or the definition of stability fails. Thus for  , the Z transform no longer exists on the unit circle.

Above we considered the case with single pole, it will give the same result with multiple pole transfer function. To summarize all poles of an LTI System should lie withing the unit circle for a stable system.