Inequality involving vector spaces

Question

Let $\{W_i\}_{i=1}^k$ be $k$ subspaces of a vector space $V$.

Prove that,

$$ \dim (W_1 + W_2 + \cdots + W_k) \leq \dim(W_1) + \dim(W_2) + \cdots + \dim(W_k)$$

I don't know how to proceed to prove this.

Any help will be appreciated.
Thanks in advance.

Answer 1

For each $i\in\{1,\ldots,k\}$, let $b_i$ be a basis of $W_i$ and let $b=\bigcup_{i=1}^kb_i$. Then$$\operatorname{span} b=W_1+W_2+\cdots+W_k$$and therefore$$\dim(W_1+W_2+\cdots+W_k)\leqslant\#b\leqslant\sum_{i=1}^kb_i=\sum_{i=1}^k\dim W_i.$$