et the sequence {xn} satisfy | Xn+1-Xn < Show that {x} is convergent. Steps to follow: ● Let m = n+ k, where k > 0 Show that ● |xn-Xn+k|≤|xn-Xn+1| + | Xn+1 - Xn+2+...+ Xn+k-1 - Xn+k| 1 + ... + 5n+k-1 Find ● ● Show that ● ● 1 + 5n 5n+1 1 1 + 57 5n+1 + + Explain why www Xn-Xntk S 1 1-k 572 |xn-Xn+k|=== 1 5n+k-1 n . Let E> 0 be given. Explain why there is N, such that < € Take m,n > N, and assume that m = n +k, k > 1. Show that |xn-Xml < E Conclude that {x} is a Cauchy sequence. Explain why you can say that (x) is convergent.

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