Someone please check my work and let me know if it's correct

Please state all definitions and theorems that you will need: Intermediate Value Theorem 5.3.6 Suppose that f: [(1, b] > R is continuous. Then 1' has the intermediate value property on [a, b] . That is, if k is any value between f(a) and f(b) [17. 8- aa) R is continuous and f([aI b]) g; Q . Let f(a) = a: and f(b) : y where a: and y are real numbers such that a: k > y , which by the law of trichotomy means that a: = k = y . Therefore, It must be a constant