Question 6.T.6: (Intermediate Value Property) Let f : [a, b] → R be continuo...
(Intermediate Value Property)
Let f : [a, b] → \mathbb{R} be continuous. If λ is a real number between f(a) and f(b), then there is a point c ∈ (a, b) such that f(c) = λ.
Assume f(a) < f(b), otherwise consider −f instead of f. Let
S = {x ∈ [a, b] : f (x) < λ}.
The set S is clearly not empty, as it includes a, and is bounded above (by b). Therefore it has a supremum which lies in [a, b]. Let c = sup S.
As we did in the proof of Theorem 6.5, we can form a sequence (x_{n}) in S which converges to c. Therefore f(x_{n}) → f(c) by the continuity of f. Since f(x_{n}) < λ for every n ∈ \mathbb{N}, we can use Theorem 5.7 to write
f(c) ≤ λ < f(b), (6.4)
which implies c < b. Now we form a sequence (t_{n}) in (c, b] which converges to c, such as
t_{n} = \min \left\{c + \frac{1}{n}, b \right\}.
Since t_{n} ∈ [a, b] and t_{n} > c = \sup S, it follows that t_{n} \notin S for any n.
Hence f(t_{n}) ≥ λ for all n ∈ \mathbb{N}, and
f(c) = \lim f(t_{n}) ≥ λ.
In view of (6.4) we must conclude that f(c) = λ.