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We define the pretangent space $\\Omega_{\\infty, \\tilde r}^{X}$ to $(X, d)$ at infinity as a metric space whose points are equivalence classes of sequences $(x_n)_{n\\in\\mathbb N}\\subset X$ which tend to infinity with the speed of $\\tilde r$. It is proved that the pretangent spaces $\\Omega_{\\infty, \\tilde r}^{X}$ are complete for every unbounded metric space $(X, d)$ and every scaling sequence $\\tilde r$. 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