Let X be a set, and define d(x,y) ={0, if x=y; 1, otherwise A sequence (xn) ⊆ X is called eventually constant if there is N ∈ N such that xm=xn; m, n > N. Show that any eventually constant sequence converges.

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A: Proof is given below Explanation:Step 1:Step 2: Step 3: Step 4:

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Q: 14. Prove that every Cauchy sequence is bounded (Theorem 1.4)

A: To prove: Every Cauchy sequence is bounded.

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A: Solution :- True Explanation :-

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.