# Bounds of a function and limits

1. Nov 8, 2007

### javi438

1. The problem statement, all variables and given/known data
A function is decreasing if f(x$$_{1}$$) > f(x$$_{2}$$) whenever x$$_{1}$$ < x$$_{2}$$, and x$$_{1}$$, x$$_{2}$$ $$\epsilon$$ $$\Re$$

a) Show that the set {f(x) : x < a} is bounded below
b) Prove that lim (as x goes to a) f(x) = glb{f(x) : x < a}
(hint: show that for any $$\epsilon$$ > 0, there exists $$\delta$$ > 0 such that f(a - $$\delta$$) < c + $$\epsilon$$

i have no idea where to starttttt ><

2. Nov 8, 2007

### HallsofIvy

You might START by stating the problem correctly. You give a definition of "decreasing function" but there doesn't appear to be a requirement that the "f" in the theorem be decreasing! Did you intend that f be decreasing? If so, make use of the definitions, since those are all you have! You know that x< a and you know that f is decreasing. "Bounded below" means "has a lower bound" which itself means that there is some number less than or equal to every number in the set. Can you make a guess at what a lower bound for {f(x)| x< a} must be? (Hint: look at f(a).)

If a set of real numbers has a lower bound, then it must have a greatest lower bound. Use the definition of "greatest lower bound" in the hint.