If I am not mistaken, the proof in the article works for √3, and your lemma 1 can be modified by noting that when x > 1, √x > 1 and also 1 < 3 < 4 so 1 < √3 < 2

If I am not mistaken, the proof in the article works for √3, and your lemma 1 can be modified by noting that when x > 1, √x > 1 and also 1 < 3 < 4 so 1 < √3 < 2, 0 < √3 - 1 < 1, and k' = k(√3 - 1) < k.