Use the Heine–Borel Theorem to prove that if \(f\) is continuous on \([a,b]\) and \(f(x)>0\) for every \(x\in[a,b]\), then there exists \(\varepsilon>0\) such that \(f(x)\ge\varepsilon\) for every \(x\in[a,b]\).
Solution
Goal: Use the Heine–Borel theorem to prove: \(f\) continuous on \([a,b]\) with \(f(x) > 0\) for all \(x\) implies \(f(x) \ge \varepsilon\) for some \(\varepsilon > 0\) and all \(x\).
1. Cover \([a,b]\) by neighborhoods where \(f\) is bounded below by a positive constant.
1.1. For each \(x_0 \in [a,b]\): since \(f(x_0) > 0\) and \(f\) is continuous at \(x_0\), choose \(\delta_{x_0} > 0\) with \(|f(x) - f(x_0)| < f(x_0)/2\) for \(|x - x_0| < \delta_{x_0}\); then \(f(x) > f(x_0)/2 > 0\) on \((x_0 - \delta_{x_0}, x_0 + \delta_{x_0}) \cap [a,b]\).Proof: \(\varepsilon\)-\(\delta\) definition of continuity with \(\varepsilon = f(x_0)/2\).
1.2. The intervals \(\{B(x_0, \delta_{x_0})\}\) form an open cover of \([a,b]\).Proof: each \(x_0\) lies in its own ball.
2. Extract a finite subcover \(B(x_1, \delta_{x_1}), \ldots, B(x_n, \delta_{x_n})\).Proof: Heine–Borel: \([a,b]\) is compact, so every open cover has a finite subcover.
3. Set \(\varepsilon := \min_i \frac{f(x_i)}{2} > 0\); then \(f(x) \ge \varepsilon\) for all \(x \in [a,b]\).Proof: every \(x \in [a,b]\) lies in some \(B(x_i, \delta_{x_i})\) by 2, where \(f(x) > f(x_i)/2 \ge \varepsilon\) by 1 and the definition of \(\varepsilon\); \(\varepsilon > 0\) since each \(f(x_i) > 0\) and the minimum is over finitely many positive numbers.
4. Q.E.D. Proof: 3 is the claim.