Prove division in quotient spaces

Question

Let $U$ and $V$ be subspaces of a vector space $W$ and suppose that $U \subset V$. The quotient space $V/U$ is a then a subspace of $W/U$ in a natural way. Prove that we have an isomorphism: $$(W/U)/(V/U) \cong W/V$$ Not sure where to get started with this, any kind of help would be greatly apprecited. Thanks in advance.

Answer 1

Consider the map$$\begin{array}{rccc}f\colon&W/U&\longrightarrow&W/V\\&w+U&\mapsto&w+V.\end{array}$$It is clearly surjective and $\ker f=V/U$. Therefore$$(W/U)/(V/U)\simeq W/V.$$