Theorem (For a converging sequence). Assume that is a continuous function and that is a sequence generated by fixed point iteration.

If  ,  then is a fixed point of .  

Proof.

If  ,  then  .  

It follows from this, and the continuity of  ,  and the relation    that

    .  

Therefore, is a fixed point of .  

Q. E. D.

(c) John H. Mathews 2004