The set of bounded elements of a locally convex algebra is characterized as the union of certain naturally defined normed subalgebras. Pseudocomplete locally convex algebras are characterized in terms of the completeness of these subalgebras. In this paper it is proved, among other results, that a convergent sequence of elements in a bounded pseudo-complete locally convex strict inductive limit algebra is locally convergent. Along the way, a theorem on $i$-bounded sets is obtained, which identifies non-normable locally convex algebras in which every bounded subset is $i$-bounded.