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If for the sequences a_n, b_n, c_n we have a_n ≤ b_n ≤ c_n for every n ∈ N and additionally lim_{n → ∞} a_n = g and lim_{n → ∞} c_n = g that lim_{n → ∞} b_n = g. Proof. Consider any ε * 0. The sequences a_n and c_n converge to g then can find number N such |a_n - g| * ε and |c_n - g| * ε. From absolute value g-ε * a_n * g+ε, g-ε * c_n * g+ε. Because a_n ≤ b_n and g-ε * a_n that g - ε * b_n. Similar b_n ≤ c_n, c_n * g+ε that b_n * g+ε. In this way get g-ε * b_n * g+ε → |b_n-g| * ε. This is equivalent that sequence b_n converge to g. End of the proof.