The club filter is the least normal tail filter

For a sequence of club-filter members choose a witnessing club inside each. Their diagonal is a club contained in the diagonal of the original members, so that diagonal belongs to the club filter. Tails are clubs, giving normality and tail containment.

Let $F$ be a proper normal tail-containing filter and $C$ a club. If $C\mathrm{\notin }F$, then $S=(\kappa \setminus C)\cap [1,\kappa )$ is positive: intersecting a positive set with a filter member preserves positivity, as every further filter intersection remains in $F$. On $S$ put $f(\alpha )=\sup (C\cap \alpha )$. At a successor this is below $\alpha$; at a nonzero limit equality would put $\alpha$ in the closed $C$. Hence $f$ is regressive.

For any $\xi$, take $c\in C$ above $\xi$. A point $\alpha >c$ has $f(\alpha )\ge c>\xi$, so the fibre of $\xi$ is bounded by $c+1$ and is disjoint from a tail in $F$. All fibres are small, contradicting positive pressing down. Thus $C\in F$, and upward closure includes the entire club filter in $F$.