Show supremum of an interval made by a continuous, increasing function

[k3k3]

Homework Statement

Let f be an increasing function defined on an open interval I and let c ϵ I. Suppose f is continuous at c.

Prove sup{f(x)|x ϵ I and x < c} = f(c)

Homework Equations

The Attempt at a Solution

Since I is an open interval and c is not able to be an end point, then {f(x)|x ϵ I and x < c}. Also, since f(c) is an upper bound on {f(x)|x ϵ I and x < c}, the supremum exists for this set.

Let sup{f(x)|x ϵ I and x < c} = β

For x < c, f(x) < f(c) since f is an increasing function, then {f(x)|x ϵ I and x < c}→f(c)

Since β is the least upper bound, then β≤f(c)

Since {f(x)|x ϵ I and x < c}→f(c), f is continuous, and β is not in the set, β=f(c).




My question is if this is correct? Did I miss how to do it entirely or does this show it properly?

 

[mighty2000]

I appreciate any help, thank you!

Hello,

Thank you for posting your solution to the forum post! Your solution looks correct and well-written. You have correctly used the definition of continuity to show that the supremum of the set is equal to f(c), and your reasoning is clear and concise. Great job!

Just one minor suggestion - you could mention in your solution that f(c) is indeed an upper bound for the set {f(x)|x ϵ I and x < c}. This will make your reasoning more complete and explicit.

Overall, your solution is well done and shows a good understanding of the concept. Keep up the good work!