Linear mapping/Composition/Fact

Let ${}K$ denote a field, and let ${}U,V,W$ denote vector spaces over ${}K$. Suppose that

$\varphi :U\longrightarrow V\,\,{\text{ and }}\,\,\psi :V\longrightarrow W$

are linear mappings.

Then also the

composition

$\psi \circ \varphi \colon U\longrightarrow W$

is a linear mapping.

Write a proof