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.