Finally, Proposition A Sample Clauses

Finally, Proposition A. 0.1 implies that H(y) = sgn(y)(|y| + κ) + R(y) where κ > 0 and R(y) ≤ O(y−2) for large |y|, which shows that H grows at most linearly at ±∞. H(0) = 0 follows from G(0) = 0. That H(y) → ±∞ as y → ±∞ follows as sgn(x)|x| = x for every x ∈ R. Finally, Hj(y) = 1 + sech2 2g(y) and so |Hj(y)| ≤ 2 for every y ∈ R.
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