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

1. Sep 15, 2012

### k3k3

1. The problem statement, all variables and given/known data
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)

2. Relevant equations

3. 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?