On closures of discrete sets

The depth of a topological space X (g(X)) is defined as the supremum of the cardinalities of closures of discrete subsets of X. Solving a problem of Martinez-Ruiz, Ramirez-Paramo and Romero-Morales, we prove that the cardinal inequality |X| <= g(X)(L)((X)) (F)((X)) holds for every Hausdorff space X, where L(X) is the Lindeloof number of X and F (X) is the supremum of the cardinalities of the free sequences in X.