I know there's a simpler way to solve this problem but could someone please check my work and let me know if it is correct

Please state all definitions and theorems that you will need: AI WH Definition 5.4.1 I noitesup Let f: D - R . We say that f is uniformly continuous on D if for every & > 0 there exists a 8 > 0 such that If(x) - f(y)| 0. Then prove that ->- for a > 0. ey 1 Show that - y for a E (0, 1 ) . Let & = 1 and a = min (6, 0.4) () What happens if y = 1 - x - ? Since e" is increasing when x > 0 , then e > e , which means > - for all x > 0 . ac Let & = 1 . Then, given any 6 > 0 , letx = min {6, 0.4} and y = 1 - x Then, if x, y E (0, 2) and (1 - x)2 -2 x - 22 la - yl = 2 - 1 - x x 1 - x we have If ( 20) - f(y)1 = y since y = 1 - 2 1 -2 1 - 1- 1+x - = 1 = E Thus, f is NOT uniformly continuous on (0, 2) by definition 5.4.1