Limits and Equilibrium Points -- A Theorem and Proof

Theorem:

Consider the discrete dynamical system

p(1) = _____

p(n + 1) = f(p(n))

and suppose that f is a continuous function. If

then L is equilibrium point.

Proof:

The proof is based on the following theorem whose proof is found elsewhere.

Suppose that f is a continuous function and that

Then

Now by this theorem we have, for the sequence

p(n + 1) = f(p(n))

But the sequence

f(p(1)), f(p(2)), f(p(3)), ...

is just

p(2), p(3), p(4), ...

That is, it is the same sequence as the original sequence but starting one term later. Thus, it has the same limit. and we see that

f(L) = L.

as we wanted to prove.