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Source. Lemma 1, printed p. 150 (published PDF).

Statement. If $M=(E,\mathcal F)$ is a finite matroid and $n\in\mathbb Z_{\ge0}$, then

$$\mathcal F_{(n)}=\{I\in\mathcal F:|I|\le n\}$$

is a matroid on $E$, with rank $r_{(n)}(A)=\min(n,r(A))$.

Proof. The family contains the empty set and is hereditary. If $I,J$ belong to it and $|I|<|J|$, ordinary matroid augmentation supplies $e\in J\setminus I$ for which $I\cup\{e\}$ is independent in $M$. Its size is at most $|J|\le n$, so it remains in $\mathcal F_{(n)}$. The augmentation characterization therefore proves the matroid property.

Every independent subset of $A$ in the truncation has size at most $\min(n,r(A))$. A base of $A$ in $M$ has $r(A)$ elements, and selecting $\min(n,r(A))$ of them attains that bound. This also covers $n=0$ and $A=\varnothing$. $\square$

The source calls the proof of Lemma 1 obvious. The argument above expands it. The source asserts, without giving the argument, that a truncation of a graphic or transversal matroid need not stay in that particular class; later uses concern general matroids.