Complemented basic sequences in Frechet spaces with finite dimensional decomposition

Let $E$ be a Frechet-Montel space and $(E_n{)}_{n\in \mathbb{N}}$ be a finite dimensional unconditional decomposition of $E$ with $\dim (E_n)\le k$ for some fixed $k\in \mathbb{N}$ and for all $n\in \mathbb{N}$. Consider a sequence $(x_n{)}_{n\in \mathbb{N}}$ formed by taking an element $x_n$ from each $E_n$ for all $n\in \mathbb{N}$. Then $(x_n{)}_{n\in \mathbb{N}}$ has a subsequence which is complemented in $E$