Contraction Mapping Theorem: Proving Continuity and Convergence of a Sequence

[kathrynag]

Homework Statement

Let f be a function defined on all of R and assume there is a constant c such that 0<c<1 and |f(x)-f(y)<c|c-y|
a) Show f is continuous on all of R
b)Pick some point y1 in R and construct the sequence (y1,f(y1),f(f(y1)),...)
In general if y_(n+1)=f(yn) show that the resulting sequence yn is a Cauchy sequence. hence we may let y=limyn
c)Prove that y is a fixed point of f and that is unique in this regard.
d) Finally prove that if x is any arbitrary point in R then the sequence (x,f(x),f(f(x)),...) converges to y defined in (b).

Homework Equations

a) want to show if |x-c|<delta then |f(x)-f(c)|<epsilon
b) A sequence is Cauchy if |an-am|<epsilon
c)want to show f(y)=y

The Attempt at a Solution

 

[arkajad]

Take a). Your "want to show if |x-c|<delta then |f(x)-f(c)|<epsilon" is not precise enough. There will always be some epsilon. In mathematics the terms and the order of terms "For any", "there exists", "such that" are of crucial importance.