Block sequences in nuclear Fréchet spaces with basis

It is shown that a block sequence in a nuclear Fréchet space with a basis has a block extension if and only if the subspace it generates is complemented. In addition, a short proof is given of the following result of Dubinsky and Robinson: a nuclear Fréchet space is isomorphic to $\omega=R^N$, $N=\{1,2,\dots\}$ if it has a basis such that any block sequence with blocks of length $\le2$ of any permutation of this basis has a block extension. It is shown that a similar result holds without considering permutations of the basis if the length of the blocks is arbitrary.