Let f be a function defined on all of R, and assume there is a constant c such that 0|f(x) -f(y) | c|x-y| for all x,y ∈ R a. Show that f is continuous on R. b. Pick some point yi₁ ∈ R and construct the sequence (y₁, f(y₁) , f(f(y₁) ) .) In general, if yₙ₊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 (i.e., f(y) =y) and that it 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.
