sLet $a_1, \ldots , a_n$ be a basis of linear space $V$ let $W \le V$ be a $k$ dimensional subspace $k \ge 1$ Show for each subset $\displaystyle a_{i_i}, \ldots a_{i_m}$ for $m>n-k$ exist non zero vertor $\beta \in W$ which is linear combination of $\displaystyle a_{i_i}, \ldots, a_{i_m}$
My try: from cond. $m>n-k$ we have at least one vector from $W$ belongs to $\displaystyle a_{i_i}, \ldots, a_{i_m}$ so we can pick this vector and we have $\beta \in \operatorname{span}( a_{i_i}, \ldots, \beta , \ldots, a_{i_m})$. Is my reasoning OK ?