Use the fact that intervals are connected to prove the intermediate value theorem.
Solution
Goal. Prove the intermediate value theorem using connectedness of intervals.
1. Let \(f: [a,b] \to \RR\) be continuous, and suppose \(f(a) < c < f(b)\) (the other case is symmetric).
setup.
2. \([a,b]\) is connected.
intervals are connected.
3. \(f([a,b])\) is connected.
the continuous image of a connected space is connected.
4. \(f([a,b])\) is a connected subset of \(\RR\), hence an interval.
the connected subsets of \(\RR\) are exactly the intervals.
5. \(f(a), f(b) \in f([a,b])\), so the interval \(f([a,b])\) contains \([f(a), f(b)]\).
an interval containing \(f(a)\) and \(f(b)\) contains everything between them.
6. Hence \(c \in [f(a), f(b)] \subseteq f([a,b])\), so there is \(x \in [a,b]\) with \(f(x) = c\).
\(c\) is between \(f(a)\) and \(f(b)\).
7. Q.E.D.
6 is the intermediate value theorem.