Sum of subspaces and span of the union

Proposition. Let $M$ and $N$ be linear subspaces of a vector space $X$. Define their sum using set addition:

Then:

1. $M+N$ is a linear subspace of $X$.
2. $M+N=\operatorname{span}(M\cup N)$.

Proof sketch. Closure of $M+N$ under addition and scalar multiplication follows from closure of $M$ and $N$ and the definitions of set sum and scalar multiples. For (2), note that $M\cup N\subset M+N$ (since $0\in M$ and $0\in N$), so the span of $M\cup N$ is contained in $M+N$. Conversely, any subspace containing $M\cup N$ contains $M+N$, hence $M+N$ is contained in the span.