Let $A\subset B$ be flats of a matroid $M$ such that $r(A)=r(B)<\infty$. Then $A = B$.

Question

I want to prove the following lemma:

Let $r$ denotes the rank.

Lemma. Let $A\subset B$ be any flats of a matroid $M$ such that $r(A)=r(B)<\infty$. Then $A = B$.

My thoughts are:

I know that $cl(A) = A$ and $cl(B) = B$ by the definition of a flat but then how this leads me to that $A=B$?

Any help will be greatly appreciated.

Answer 1

Let $x\in B$, and suppose $x\not \in A$, then $r(A\cup \{x\})=r(A)+1\leq r(B)$(the first equality comes from definition of flat). This contradicts the hyphothesis. This implies that $B\subseteq A$.