 Local compactness of a product with finitely many noncompact factors

A product topology on locally compact factors is locally compact if all but finitely many factors are compact. At a point, choose a compact neighborhood in each of the finitely many exceptional factors and take the full compact space in every other coordinate. Tychonoff's theorem makes their product compact, and the product contains an open basic neighborhood because only finitely many coordinates were restricted. In contrast ${\mathbb{R}}^{\mathbb{N}}$ is not locally compact: a basic neighborhood has a free coordinate projecting onto all of the noncompact real line, so no compact set can contain it.