Prove: Suppose that {x n } is a convergentsequence; that is, x n -->x 0 asn-->.

Question

Prove: Suppose that {xn} is a convergentsequence; that is, xn-->x0 asn-->. If {yn} is an infinite subsequence of{xn}, then yn-->x0 as n--> **Please show all work. Prove: Suppose that {xn} is a convergentsequence; that is, xn-->x0 asn-->. If {yn} is an infinite subsequence of{xn}, then yn-->x0 as n--> **Please show all work.

Explanation / Answer

x_n converges so it satisfies the cauchy condition. In particular, y_n will satisfy the cauchy condition too, so that y_n must converge to something. Say it converged to y_0. If x_0 - y_0 > 0, then we can set epislon = (x_0 - y_0)/2. Then for large n, the y_n will be close to y_0 hence will be epsilon away from x_0 contradicting the assumption that x_n converged to x_0. Fill in the details =)

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