# a fixed point theorem

A hiker starts to climb up from base B to the summit S on sat 6am one day,spends the night at S and starts to climb down at 6am the next day.Prove that there is a point on the path B-S (there is only one path connecting B-S) which hiker crosses at the same time both days.

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Do you know the intermediate value theorem? It states that if $f:[a,b] \rightarrow \mathbb{R}$ is a continuous function satisfying $f(a) = -1, f(b)=1$, then for any $x \in [-1,1]$ , there exists $c \in [a, b]$ such that $f(c) = x$.
Apply this to a suitable function describing your hiker's movement, and let $x=0$.