  Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite
positive dimension. We consider a pair of linear transformations $A:V \to V$
and $A^*:V \to V$ that satisfy the following conditions: (i) each of $A,A^*$ is
diagonalizable; (ii) there exists an ordering $\lbrace V_i\rbrace_{i=0}^d$ of
the eigenspaces of $A$ such that $A^* V_i \subseteq V_{i-1} + V_{i} + V_{i+1}$
for $0 \leq i \leq d$, where $V_{-1}=0$ and $V_{d+1}=0$; (iii) there exists an
ordering $\lbrace V^*_i\rbrace_{i=0}^\delta$ of the eigenspaces of $A^*$ such
that $A V^*_i \subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$ for
  $0 \leq i \leq \delta$, where $V^*_{-1}=0$ and $V^*_{\delta+1}=0$; (iv) there
is no subspace $W$ of $V$ such that $AW \subseteq W$, $A^* W \subseteq W$, $W
\neq 0$, $W \neq V$. We call such a pair a {\it tridiagonal pair} on $V$. It is
known that $d=\delta$ and for $0 \leq i \leq d$ the dimensions of $V_i$,
$V^*_i$, $V_{d-i}$, $V^*_{d-i}$ coincide; we denote this common dimension by
$\rho_i$. In this paper we prove that $\rho_i \leq \rho_0 \binom{d}{i}$ for $0
\leq i \leq d$. It is already known that $\rho_0=1$ if $\K$ is algebraically
closed.
