Time shifting property of z transform

Before we learn about time shifting property first let we learn about what is z transform system in digital signal processing subject.

Statement : 

              Z

If  x(n) ↔ X (Z)

                      Z

Then x (n-k) ↔ Z ^-k  X (Z)

Proof :

                                 -∞

So here Z{x(n)} = X  (Z) = ∑     x (n)  Z-n

                                 +∞

Z { x(n-k) }  can be written as,

                    -∞

Z{x(n-k)} = ∑     x (n-k)  Z-n

                     +∞

Now  Z-n can be  written as  Z-n  =   Z-(n-k) Z-k  thus the equation 

                    -∞

Z{x(n-k)} = ∑     x (n-k)  Z-(n-k) Z-k

                     +∞

Since the limits of summation are in terms of  'n'  we can take Z-k outside the summation sign.

                             -∞

so Z{x(n-k)} = Z-k  ∑     x (n-k)  Z-(n-k) 

                             +∞

Now put n-k =m on R.H.S 

At  n = -∞, -∞-k = m → m =  -∞

At n = ∞, ∞-k = m  → m =  ∞

                              ∞

Z{x(n-k)} = Z-k  ∑     x (m)  Z-(m) 

                          m = -∞ 

                  

Comparing Equation : 

                           

Z{x(n-k)} = Z-k   X (Z)

= x (n-k) ↔ Z-k   X (Z)

Similarly, we can write 

= x (n+k) ↔ Z+k   X (Z)