  In "All p-adic reductive groups are tame" Bernstein proved that for a
reductive group G over a local non-archimedean field F and a compact open
subgroup K of G there exists a uniform bound N(G,K) such that for every
irreducible, smooth, and admissible representation V of G the dimension of the
subspace of K-fixed vectors in V is bounded by N(G,K). In this note I repeat
the proof of Bernstein and give my proof to one of the two main lemmas. The new
proof of this lemma gives a new, sharper bound for the constant N(G,K).
