(a) The projection is an open continuous map, therefore, we may use the next exercise to argue that all $X_\alpha$ are locally compact. A compact subspace of the product containing an open set has all but finitely many projections equal to the whole corresponding space, since the projection is continuous, these spaces must be compact.(b) Assuming the Tychonoff lemma all we need to prove is that the product of two locally compact spaces is locally compact. For any $x\times y$ find the corresponding compact subsets and neighborhoods in both spaces and take their products.