let X n and Y n be two sequences in theReal line. Let Z n be the sequence given

Question

let Xn and Yn be two sequences in theReal line. Let Zn be the sequence given by(x1,y1,x2,y2,x3,y3,....) Show that Zn converges if and only if Xnand Yn both have the same limit in the Real line. let Xn and Yn be two sequences in theReal line. Let Zn be the sequence given by(x1,y1,x2,y2,x3,y3,....) Show that Zn converges if and only if Xnand Yn both have the same limit in the Real line.

Explanation / Answer

One way is easy. If a sequence converges to a limit L, thenevery subsequence converges to L as well. As Xn and Yn areboth subsequences of Zn, if Zn converges, so must Xn and Yn. Conversely, suppose Xn and Yn both converge to L. Then givenE > 0, there is an N1 and N2 such that n > N1 implies |Xn -L| N2 implies |Yn - L| 2*Max(N1, N2). Then for n > N, if we look at thedefinition of Zn, we are beyond both N1 and N2 in our subscripts onthe Xn, Yn. So |Zn - L| N. So Znconverges to L.

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