A possible application of the intermediate value theorem

Question

Let $f : [a,b] \rightarrow \mathbb{R}$ be continuous and $f(x) > 0$ for all $x \in [a,b]$. Show that there exists $\delta > 0$ such that $f(x) \geq \delta$ for all $x \in [a,b]$.

Answer 1

If no such $\delta$ exists then there is a sequence $(x_n)$ in $[a,b]$ such that $0 < f(x_n) < 1/n$ and thus $f(x_n) \to 0$.

The sequence $(x_n)$ is bounded and by Bolzano-Weierstrass has a convergent subsequence $(x_{n_k})$. Since $[a,b]$ is closed there is a point $c \in [a,b]$ such that $x_{n_k} \to c$.

Can you finish by showing that $0 = \lim_{k \to \infty}f(x_{n_k}) = f(c)$ -- a contradiction?