“…v s,k − z 2 ≤ p s,k − z 2 − 2α k (f (ž) − f (z)) + 1 4η p s,k −ž 2 + (1 + 4η)α 2 k L 2 f .Substituting this inequality in relation (50), we obtainz s,k − z 2 ≤ p s,k − z 2 − 2α k (f (ž) − f (z)) + 2β s,k δ s,k v s,k − z + 2β s,k |ν s,k | + 1 4η p s,k −ž 2 + A η,τ α 2 k L 2 f − τ − 1 τ (L h + D) 2 h 2 + (p s,k ; D s,k , ω s,k ),where A η,τ = 1+4η +τ . From relation(15), we have v s,k − z ≤ C z . Using this and the upper estimate for β s,k in(21)for bounding the three error terms completes the proof.Proof of Lemma V.8.…”