Banach F.P. Th Theorem 3.18: The Contraction Lemma Let (X, d) be a complete metric space and f: X → X a contractive function. Then there is exactly one point x in X for which f(x) = x. x is called a...

Extracted text: Banach F.P. Th Theorem 3.18: The Contraction Lemma Let (X, d) be a complete metric space and f: X → X a contractive function. Then there is exactly one point x in X for which f(x) = x. x is called a fixed point of f (any). Proof: To show the existence of such a point, choose a point x, in X and define x2 = f(x1), x3 = f(x2), ..., Xn = f(xp-1),. n2 2. The fact that f is a contractive function insures that {xn} is a Cauchy sequence. By completeness, this sequence has a limit x in X. A V w- wnOSE m Thus f(x) = x. To show the required uniqueness property, suppose that y is a second point satisfying f(y) = y. Then d(x, y) = d(f(x), f(y)) s ad(x, y). Since a < 1, this relation cannot hold unless d(x, y) = 0 and x = y. show dian, tnt ) * d(x,%2) use seguence definition exoma hwl showw d(xn, xntk t.. xuti l use o to bound the right-side of o (& & shows hhat { zu } is cauchy (we a 1,="" this="" relation="" cannot="" hold="" unless="" d(x,="" y)="0" and="" x="y." show="" dian,="" tnt="" )="" *="" d(x,%2)="" use="" seguence="" definition="" exoma="" hwl="" showw="" d(xn,="" xntk="" t..="" xuti="" l="" use="" o="" to="" bound="" the="" right-side="" of="" o="" (&="" &="" shows="" hhat="" {="" zu="" }="" is="" cauchy="" (we="">