3. (30 points) Let {sn} be a sequence of real numbers and define the arithmetic mean...

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3. (30 points) Let {sn} be a sequence of real numbers and define the arithmetic mean on by $1+ + Sn On := (a) Prove rigorously that if " n = 1, 2, ... n lim Sn = 0, then sn n→∞ lim on = 0. n→∞ (b) Using (a), show that if lim sn = s, then lim on = s. n→∞ n→∞ (c) Is the converse to (b) true? Explain. 3. (30 points) Let {sn} be a sequence of real numbers and define the arithmetic mean on by $1+ + Sn On := (a) Prove rigorously that if " n = 1, 2, ... n lim Sn = 0, then sn n→∞ lim on = 0. n→∞ (b) Using (a), show that if lim sn = s, then lim on = s. n→∞ n→∞ (c) Is the converse to (b) true? Explain.