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Let S be the collection of all sequences of real numbers and define a relation on...

Let S be the collection of all sequences of real numbers and define a relation on S by {xn} ∼ {yn} if and only if {xn − yn} converges to 0.

a) Prove that ∼ is an equivalence relation on S.

b) What happens if ∼ is defined by {xn} ∼ {yn} if and only if {xn + yn} converges to 0?

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