  There is another proof using the elementary amalgamation theorem and the elementary chain theorem. One can show that the structure A has a proper elementary extension A′.  (There is a proof of this using the compactness theorem and the diagram lemma — see 3.1 and 3.2 below; another proof is by ultrapowers — see 4.1 below.)  Now use A′ and again A′ for the structures B and C in the elementary amalgamation theorem. Then D as in the theorem is an elementary extension of A, and by (ii) in the theorem, it must contain elements that are not in A′, so that it is a proper elementary extension. Repeat to get a proper elementary extension of D, and so on until you have an infinite elementary chain. Use the elementary chain theorem to find an elementary extension of A that sits on top of this chain. Keep repeating these moves until you have an elementary extension of A that has cardinality at least λ. Then if necessary use the downward Löwenheim-Skolem theorem to pull the cardinality down to exactly λ. This kind of argument is very common in first-order model theory.  By careful choice of the amalgams at the steps in the construction, we can often ensure that the top structure has further properties that we might want (such as saturation, see 4.2 below).