Is b in the span of {w_1, .. , w_p}? Suppose b is in the span of { v_1 , ... , v_n }, and that each v_i is in the span of { w_1, ... , w_p }. Is b, then, in the span of { w_1, ... , w_p } ? If not, how could you modify the proposition so that it is true?
Not sure at all how to approach this.

If $b$ is in the span of $\{v_1, \dots, v_n \}$, then
$$b = \sum_{i=1}^n \alpha_i v_i,\tag{1}$$
for some $\alpha_i$. Similarly, if each $v_i$ is in the span of $\{w_1, \dots, w_p\}$, then
$$v_i = \sum_{j=1}^p \beta_{ij} w_j,\tag{2}$$
for some $\beta_{ij}$. Now, substitute $v_i$ in $(1)$ with what you have in $(2)$. Now, is $b$ in the span of $\{w_1, \dots, w_p\}$?
