I am studying the book cited, particularly this Theorem:
The space $B_0(\mathcal{N},\mathcal{B})$ of the compact operators (including those that have finite post) is Banach.
The hypothesis of $\mathcal{B}$ being complete cannot be removed and for the study the author build a counterexample using $B(\mathcal{B_1},\mathcal{N_1})$, where
$$\mathcal{B_1}=C^1[0,1]$$ with norm $$|||\psi|||=||\psi||_\infty+||\psi'||_\infty$$
and
$$\mathcal{N_1}=C^1[0,1]$$ with norm $$||\psi||_\infty$$.
At certain point the author states that, for $|||\psi|||\leq 1$, we have $$|\psi(t)-\psi(s)|\leq|t-s|$$ for all $t,s\in[0,1]$.
I could not understand why.
Many thanks.
==========
OLIVEIRA, César de. Introdução à Análise Funcional. Rio de Janeiro: Impa, 2012. (Projeto Euclides).