A finite-dimensional vector space over an infinite field is not a finite union of proper subspaces

For $m=0$, the union is empty and cannot equal the nonempty set $V$, even when $V$ is the zero space.

Assume the assertion for families of fewer than $m>0$ proper subspaces, in every finite-dimensional vector space over $F$.

If $m=1$, choose $b\in V\setminus W_0$ and the result is immediate. For $m>1$, remove any $W_i$ contained in another member; if this shortens the family, the induction hypothesis applies. Otherwise every $W_{m-1}\cap W_i$ is a proper subspace of $W_{m-1}$, so the induction hypothesis inside $W_{m-1}$ gives a nonzero $a\in W_{m-1}$ lying in none of the earlier $W_i$. Choose $b\in V\setminus W_{m-1}$.

In the unresolved case $m>1$, on the affine line $\{b+ta:t\in F\}$ each $W_i$ contains at most one point: two such points would have difference a nonzero scalar multiple of $a$, putting $a$ in $W_i$ for $i<m-1$, while any point in $W_{m-1}$ would put $b$ there.

For $m>1$ the union therefore meets the line in at most a finite set of points by [L3], whereas $t\mapsto b+ta$ is injective and $F$ is infinite. Some point of the line lies outside every $W_i$; together with the $m=1$ conclusion in step 2.1, this completes the induction.