Consider a function f : R--> R, and assume that there is a c is in (0, 1) so that
|f(x) - f(y)|<= c|x -y|
for all x, y in R.
(a) Show that f is continuous on R.
(b) Given a point y1 in R define a sequence by yn+1 = f(yn). Prove that yn is a Cauchy sequence
(and therefore convergent).
(c) Let y = lim yn. Prove that f(y) = y.
(d) Prove that the limit y is independent of the choice of y1, i.e. any sequence defined by choosing an x1 and defining xn+1 = f(xn) converges to the y from part (c).
The Attempt at a Solution
a) Well this is what I tried, I'm not too sure if it is right:
Let epsilon>0 and choose delta=epsilon/c so that
0<|x-y|<delta implies that
Since this holds for all x, y in R, that means that the function is continuous on R
b) Not too sure how to do this, but I know that for epsilon > 0 you need to pick an N in N so that for m, n >= N:
c)No idea what to do,
by definition of a limit of a sequence: for all epsilon>0
d) Need part c to do this one....