Let (M,d) be a metric space and assume f:M--->M is a contraction mapping...

so f is also Lipschitz with a constant <1. Assume a sequence Xn is in M and is defined by Xn+1=f(Xn).
How do you prove that Xn is a contractive sequence?

You don't mention how the sequence starts, so I'll assume that x_0 is just some unspecified element in M.

Call the Lipschitz constant α, so 0 < α < 1. Let n be any positive integer. Note that

|x_(n+2) - x_(n+1)| = |f(x_(n+1)) - f(x_n)| ≤ α |x_(n+1) - x_n|.

Hence {x_n} is contractive.