Prove that $\operatorname{rank}(A) + \operatorname{rank}(B) \ge \operatorname{rank}(A + B)$

Question

Let $A,B$ be matrices $m\times n$ $(A, B \in M_{m\times n}(R))$. How can we prove that $$\operatorname{rank}(A+ B) \le \operatorname{rank}(A) + \operatorname{rank}(B)\ ?$$

Answer 1

Hint: Let $e_1,\ldots,e_n$ be the standard basis of $\Bbb R^n$, then the spanned subspaces satisfy: $$\langle \ldots, (A+B)e_i,\ldots \rangle \subseteq \langle \ldots, Ae_i,\ldots \rangle +\langle \ldots,Be_i,\ldots\rangle .$$