Show that there is a basis outside the intersection of proper subspaces

Show that there is a basis outside the intersection of proper subspaces

Let $U_1,...,U_m$ be proper subspaces of an $n$-dimensional vector space
$V$. Show that there is a basis $\lbrace v_1,...,v_n \rbrace$ of $V$
outside the set $U_1 \cap...\cap U_m$.
Attempt: I understand that since the subspace are proper, their dimension
is less than n. So at most the intersection could have dimension of $n-1$.
But how to show that an entire basis could be obtained?