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Let H = ( V , E ) be an s -uniform hypergraph of order n and k ≥ 0 be an integer. A k -independent set S ⊆ V is a set of vertices such that the maximum degree in the hypergraph induced by S is at most k . The maximum cardinality of a k-independent set of H is denoted by α k ( H ) . In this paper, we give a lower bound of α k ( H ) for H in terms of its maximum degree. Furthermore, we prove for all k ≥ 0 that α k ( H ) ≥ s ( k + 1 ) n 2 d + s ( k + 1 ) , where d is the average degree of H .