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.
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)##
∴ 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.