Prove f has maximum value in R

[lep11]

Homework Statement

Let f: ℝ→ℝ be continuous function such that f(x) approaches 0 when x---> -∞ and f(0)=2. In addition, f is decreasing when x≥3. Prove f has a maximum (greatest value) in ℝ.

Homework Equations

Theorem: If function f is continuous on a closed interval [a, b], then f has maximum and minimum in [a, b].

3. The Attempt at a Solution
Because f(x) approaches 0 when x ---> -∞ then there exists a<0 such that a<0 ⇒ |f(x)-0|=|f(x)|<2. So a<0 ⇒ f(x)<2. Because f is continuous in ℝ, it's also continuous on a closed interval [a,3]⊂ℝ and there's the theorem in our lecture notes that says now ∃x₀∈[a,3] such that f(x)≤f(x₀) ∀x∈[a,3]. Since a is negative, 0∈[a,3] and therefore f(x₀)≥f(0)=2. Because f is decreasing when x≥3, f(x)≤f(x₀), when x>x₀≥3.

The idea is to divide ℝ into intervals so that we get closed interval to apply the theorem.

In a summary:
$a<0 ⇒ f(x)<2≤f(x_0)$
$x∈[a,3] ⇒f(x)≤f(x_0)$
$x≥3 ⇒f(x)≤f(x_0)$

∴ f has greatest value/maxima in ℝ.

Is this proof correct and rigorious enough? I think I have made a mistake in the end when it comes to the fact that f is decreasing when x≥3.

 

[LCKurtz]

I think you have the right idea. I would change your last inequality to$$
x\ge 3 \implies f(x) \le f(3) \le f(x_0)$$to make it explicit where you use the fact that $f$ is decreasing on $[3,\infty)$.