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The sequence approaches some limit if converge to some number g. lim_{n → ∞} a_n = g. The number g is the limit of sequence a_n if for every small ε * 0, there exists number M such that, for all terms a_n with index n * M we have |a_n - g| * ε. This inequality mean that distance on a number line between the term a_n and g is less than ε. The sequence a_n approaches limit g if for all ε * 0 can find such number M that infinity amount of terms a_{M+1}, a_{M+2}, ... is inside the interval (g - ε, g + ε). Outside this interval there are finite amount of terms a_1, a_2, a_3, ...., a_M. That is always for every small ε * 0. Notation lim is abbreviation of the word limes, which in latin mean limit. Another notations: a_n → g when n → ∞.