Originally Posted by
derek walcott Space X - graph of f(x) = sin(1/x) on (0,1]
X = {(x,y): 0<x<=1, y = sin(1/x)}
Prove:
a) X is not complete
b) Every contraction h:X-->X has a fixed point
Hints for b given by my professor:
1.) X={(x,y) are elements in X: x = delta} U {(x,y) are elements in X: x >= delta}
2.) If delta is sufficiently small then diam(h(x1))<2; so h(x1) is an arc
3.) h(x2) is an arc (COMPACTNESS)
4.) therefore h(x) bounded away from the y-axis
5.) therefore h has a fixed point
Any help would be useful on any parts of these problems. He said proving part a would be simple but then he said part b would be long. Again, any help would be useful. I will post what I have for part a in a little bit ... still working on it. Thank you for reading.