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Let (x_n) be a sequence in R. Are these true or false. If every subsequence of...

Let (x_n) be a sequence in R. Are these true or false.

If every subsequence of (x_n) is Cauchy then (x_n) is Cauchy.

If (x_n) is Cauchy and (y_n) is a bounded sequence, then (x_n y_n) is Cauchy.

If ( |x_n| ) is Cauchy, then (x_n) is Cauchy.

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