Problems with an inequality involving integrals

Question

I consider $(X,M,\mu)$ a measure space. Let be $k=k(x,y)$ and$ k_n=k_n(x,y)\in L^2(X\times X,M\times M, \mu \times \mu)$.

Let be the operator $K$ defined in this way:

$(Kf)(x)=\int_{X} k(x,y) f(y) d\mu(y)$ with $f\in L^2(X,M,\mu)$

Then I consider the operators:

$(Kf_n)(x)=\int_{X} k_n(x,y) f(y) d\mu(y)$ with $f\in L^2(X,M,\mu)$

My question I don't understand why I can write:

$|(K-K_n)f(x)|^2\leq \int_X |k(x,y)-k_n(x,y)|d\mu(y)\; \int_X|k(x,y)-k_n(x,y)||f(y)|^2 d\mu(y)$

I really don't understand why I can write this inequality. Thanks for the help!

Answer 1

$|(K-K_n)f(x)|^2= \left(\int \sqrt{|k(x,y) -k_n (x,y)|}\cdot \sqrt{|k(x,y) -k_n (x,y) | }|f(y) |\mu (dy )\right)^2\\ \leq \int_X |k(x,y)-k_n(x,y)|\mu(dy)\; \int_X|k(x,y)-k_n(x,y)||f(y)|^2\mu (dy)$

By Cauchy-Buniakowski-Schwarz inequality for integrals.