Vector space/Linear subspace/Dimensions/Compare/Fact

Let ${}V$ denote a finite-dimensional vector space over a field ${}K$. Let ${}U\subseteq V$ denote a linear subspace.

Then ${}U$ is also finite-dimensional, and the estimate

${}\dim _{K}{\left(U\right)}\leq \dim _{K}{\left(V\right)}\,$

holds.

Proof, Write another proof