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Let X be an integral projective variety. Set n:= dim X. Let e(X) ≥ 2n+1 be the embedding dimension of X (we may take e(X)=2n+1 if X is smooth). Fix integers δ and r ≥ e(X). We prove the existence of many embeddings j:X↪Pr such that deg(X) ≥ δ and every point of Pr is spanned by a low degree smoothable zero-dimensional subscheme of X.