5. Let (xn) be a sequence of real numbers. Determine whether the following statements are always...

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5. Let (xn) be a sequence of real numbers. Determine whether the following statements are always true, sometimes true or never true. If true, prove the theorem using the definition of a convergent sequence. If sometimes true, give an example for which the statement is true and an example for which the statement is false. If the statement is never true, justify your answer. (a) If (xn) converges to a limit L, then (1 xn 1) converges to |L ]. (b) If (| Xn |) converges, then (xn) converges. (c) If (xn) converges to 0, then the set {n eN| Xn = 0} is infinite. 5. Let (xn) be a sequence of real numbers. Determine whether the following statements are always true, sometimes true or never true. If true, prove the theorem using the definition of a convergent sequence. If sometimes true, give an example for which the statement is true and an example for which the statement is false. If the statement is never true, justify your answer. (a) If (xn) converges to a limit L, then (1 xn 1) converges to |L ]. (b) If (| Xn |) converges, then (xn) converges. (c) If (xn) converges to 0, then the set {n eN| Xn = 0} is infinite.