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Zngg909

Mathematics

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 hello


I need to prove 15.I, and 15.M  and 15.N 

the questions are in the attachment.

thanks

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15. SUBSEQUENCES AND COMBINATIONS 103 -- (c) xn 2n 3n²+1 2n²+3 (d) xn = 3n2+1' (e) Xn = n²-n, (f) Xn = sin n. 15.D. If X and Y are sequences in Rº and if X+Y converges, do X and Y converge and have lim (X+Y)= lim X + lim Y? 15.E. If X and Y are sequences in R’ and if X · Y converges, do X and Y converge and have lim X · Y = (lim X) · (lim Y)? 15.F. If X = (xn) is a positive sequence which converges to x, then (7x) con- verges to Vx. (Hint: Vxn – Vx= (xon – x)/(Vx+Vx) when x70.) 15.G. If X = (xn) is a sequence of real numbers such that Y=(xx%) converges to 0, then does X converge to 0? 15.H. If xn = Vn+1-Vn, do the sequences X = (xn) and Y=(Vn xn) converge? 15.1. Let (xn) be a sequence in R’ such that the subsequences (x2n) and (x2n+1) converge to xe R'. Prove that (xn) converges to x. 15.J. Let (xn) and (yn) be sequences in R such that lim (xn) #0 and lim (Xnyn) exists. Prove that lim (yn) exists. 15.K. Does Exercise 15.J remain true in R?? 15.L. If 0
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The boundary point can either be in A or not in A. If x is in the boundary of A, then any
open set with x mu...


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