Equivalent Statements for a Discrete-Time System

Question

I have a discrete-time system x_(k+1) = A*x_k, x(0) = x_0 where A is in n x n dimensional space and is a real constant matrix. How do I show that the following statements are equivalent?

1. All eigenvalues of A are located on the open unit disc
2. For all x_0 in n dimensional space, the sum of ||x_k||^2 from k = 0 to infinity is less than positive infinity
3. The Lyapunov equation
4. For all x_0 in n dimensional space, the limit as k approaches infinity of x_k = 0

Any help with this problem would be greatly appreciated!