Almost disjointness number ($\mathfrak{a}$)

The least size of an infinite maximal almost disjoint family on omega. Zorn lemma extends an infinite disjoint family to a maximal one; thus $\mathfrak{a}\le 2^{{\aleph }_0}$. The bounding-to-almost-disjointness inequality supplies $\mathfrak{b}\le \mathfrak{a}$.

For an infinite almost disjoint family on omega of size less than the bounding number, select countably many members and remove their intersections with earlier selected members to obtain infinite disjoint sets $C_n$. Bound, eventually and simultaneously, the finite intersections of each member with $C_n$. Choose one point of each $C_n$ above its bound. The resulting infinite set is almost disjoint from every original member, proving the family is not maximal. A selected member has one exceptional infinite intersection, which is ignored in its bounding function.