Let $X$ be a Banach space and $1\leq p<\infty$. My question is: Are the following two statements equivalent:
(1). Every normalized weakly $p$-summable sequence in $X$ contains a basic subsequence $(x_{n})_{n}$ that is $C$-dominated by the unit vector basis $(e_{n})_{n}$ of $c_{0}$ for some $C<\infty$, that is, for all scalars $a_{1},a_{2},...,a_{n}$ and for all $n$, one has $\|\sum_{k=1}^{n}a_{k}x_{k}\|\leq C\sup_{1\leq k\leq n}|a_{k}|$;
(2). There exists a constant $\lambda>0$ such that every normalized weakly $p$-summable sequence in $X$ contains a basic subsequence $(x_{n})_{n}$ that is $\lambda$-dominated by the unit vector basis $(e_{n})_{n}$ of $c_{0}$.
