Basic Analysis I — Real Analysis · Problem 2.1.16

Sequences and Series — Jiri Lebl

Mathematics — Real Analysis · subsequences

Problem

Let {xn}n=1∞\{x_n\}_{n=1}^\infty be a sequence. Suppose there are two convergent subsequences {xni}i=1∞\{x_{n_i}\}_{i=1}^\infty and {xmi}i=1∞\{x_{m_i}\}_{i=1}^\infty. Suppose lim⁡i→∞xni=aandlim⁡i→∞xmi=b,\lim_{i\to\infty} x_{n_i} = a \quad \text{and} \quad \lim_{i\to\infty} x_{m_i} = b, where a≠ba \neq b. Prove that {xn}n=1∞\{x_n\}_{n=1}^\infty is not convergent, without using Proposition 2.1.17.

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