Let $A$ be an affine space over a vector space $M$ and $X$ be a nonempty subset of $A$.
How do I prove that $\langle\{a-b\in M: a,b\in X\}\rangle$ is the direction of $\langle X\rangle$?
($\langle X \rangle$ is defined as the intersection of all affine subspaces containing $X$)
Here's what I tried. Let $I$ be the set of affine subspaces of $A$ containing $X$. I have proven that the direction of $\langle X \rangle$ is the intersection of all directions of affine subspaces in $I$. Using this, I have shown that $\langle\{a-b\in M: a,b\in X\}\rangle$ is contained in the direction of $\langle X\rangle$. However, the other direction seems tough. How do I prove this? Thank you in advance