Let $(M,\omega)$ be a symplectic manifold. One can define its **minimal Chern number** $N_M$ as:
$$
N_M := \text{inf} \lbrace k > 0 \ |\ \exists A \in H_2(M; \mathbb{Z}), \langle c_1, A \rangle = k \rbrace,
$$
or alternatively, as the positive generator of $\langle c_1, H_2(M; \mathbb{Z}) \rangle$, where $c_1 \in H^2(M; \mathbb{Z})$ is the first Chern class of the symplectic manifold $(M, \omega)$, and $\langle ., . \rangle$ is the natural pairing between cohomology and homology groups.

On the other hand, one has the cuplength $cl(M)$ of $M$, defined as the minimal positive integer $k$ such that any cup-product $a_1 \cup ... \cup a_k$ of cohomology classes $a_j \in H^*(M; \mathbb{Z})$ of degree greater or equal to $1$ vanishes.

I have seen in several papers (for instance in the introduction of the paper "a fixed point theorem for toric manifolds", by A. Givental), that $$ N_M \leq cl(M), $$ with equality only if $M = \mathbb{C} P^n$ endowed with the Fubini-Study form, in which case both quantities equal $n+1$.

However, I have never seen a proof of this statement. Does someone have an idea of how to prove this fact?

Thank you in advance!