Existence of a root

Question

Let $f:[a,b] \rightarrow \Bbb R$ continuous, such that for every $x$ there is a $y$ such as that $|f(y)|\leq|f(x)|/2$.

Show there exists a $\xi$ such that $f(\xi)=0$

Answer 1

Since $|f|$ is continuous, it attains its minimum $m=|f(\xi)|≥0$, by assumption there must be some $y$ with $$m≤|f(y)|\leq\frac{|f(\xi)|}{2}=\frac{m}{2}$$ that is $m\leq0$, or $m=0$, and we are done.

Answer 2

Hint Pick some $x_0 \in [a,b]$. Prove that there exists some $x_n$ such that $$|f(x_n) | < \frac{|f(x_0)|}{2^n}$$

Now, use the fact that $x_n \in [a,b]$ must have a convergent subsequence.