We consider the following activation process: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it has at least active neighbors ( is identical for all vertices of the graph). Our goal is to find a set of minimum size whose activation will result in the entire graph being activated. Call such a set contagious. We give new upper bounds for the size of contagious sets in terms of the degree sequence of the graph. In particular, we prove that if is an undirected graph then the size of a contagious set is bounded by (where is the degree of ). Our proof techniques lead to a new proof for a known theorem regarding induced -degenerate subgraphs in arbitrary graphs.