In this post the author states a discrete version of the Intermediate Value Theorem as follows:
For integers $ a < b $, let $ f $ be a function from the integers in $ [a, b] $ to $ \mathbb{Z} $ that satisfies the property, $ |f(i + 1) - f(i)| \leq 1 $ for all $ i $. If $ f(a) < 0 < f(b) $, then there exists an integer $ c \in (a, b) $ such that $ f(c) = 0 $.
And the proof for this theorem was given as follows:
Proof. Let $ S = \{x \in \mathbb{Z} \cap [a, b] : f(x) < 0\} $ and let $ c = \max S + 1 $. We claim that $ f(c) = 0 $. Say $ f(c) < 0 $. Then $ c \in S $. This contradicts the fact that $ c - 1 $ is an upper bound on $ S $. Say $ f(c) > 0 $. This implies that $ f(c - 1) \geq 0 $ (by "continuity"), which contradicts the fact that $ c - 1 \in S $. $ \blacksquare $
However, it seems to me that this proof does not show $ c \in (a, b) $. What makes it obvious that $ c $ belongs to $ (a, b) $?