A topological space $X$ is sequential if it is true that an any map whose domain is $X$ is continuous iff it is sequentially continuous.
For a projective countable spectrum of sequential topological spaces (i.e. the sequence is connected with a system of continuous maps satisfying a commuatative-diagramm condition) the limit may fail to be sequential even if all steps $X_{n}$ are Fr\'{e}chet-Uryson-spaces (see Engelking/ General Topology).
But how is the situation for projective sequences of sequential LCVS (with the embeddings as connecting maps), i.e. for a sequence $(E_{n})_{n \in \mathbb N }$ such that $E_{n+1}$ embedds continuously into $E_{n}$? Clearly, if a LCVS is sequential, then it must be bornological. Bornologicity of $\varprojlim X_{n}$ is for some situation equivalent to the vanishing of $Proj^{1}(X_{n})$ (see for example the book by Wengenroth).
Is there any general theorem which guarantees that the projective limit is sequential if the all the steps are so?