We generalize the so-called Weighted König Lemma, due to Máté, for a submultiplicative function on a subset of the union , where is a set and is the Cartesian product of copies of . Instead of a combinatorial argument as done by Máté, our proof uses Tychonoff's compactness theorem to show the existence of a König chain for a submultiplicative function. As a consequence, we obtain an extension of the Daubechies-Lagarias theorem concerning a finite set of matrices with right convergent products: Here we replace matrices by Banach algebra elements, and we substitute compactness for finiteness of . The last result yields new generalizations of the Kelisky-Rivlin theorem on iterates of the Bernstein operators on the Banach space