Let (X, d) be a metric space. Define dˆ : X × X → R, by: ˆ
d(x, y) = min{1, d(x, y)}. (a) Prove that dˆ is a bounded metric on X. (b) Use part (a) to prove that for ε > 0 there exists a bounded metric dˆ on X such that for all ˆ
x,y∈X we have d(x,y)<1⇒d(x,y)<ε.

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.