$\sup$ norm of a function

Question

The following is an example of Murphy's C*-algebras and operator theory:

I do not know how he concludes $$\int_0^1 |k(s,t) - k(s',t)||f(t)| dt \leq \sup|k(s,t) - k(s',t)|||f||_\infty$$

Please help me. Thanks for your help.

Answer 1

$$\int_0^1 \underbrace{|k(s,t)-k(s',t)|}_{\leq \sup_{r \in [0,1]} |k(s,r)-k(s',r)|} \underbrace{|f(t)|}_{\leq \sup_{r \in [0,1]} |f(r)|} \, dt \leq \sup_{r \in [0,1]} |k(s,r)-k(s',r)| \cdot \|f\|_{\infty} \cdot \int_0^1 \, dt.$$