Answer to Question #158012 in Real Analysis for Cypress

Question

Find limit superior and limit inferior for the sequence (an)n∈N=((1/n)+(-1)^n)n∈N

Answer 1

$\{a_n\}=\{\frac{1}{n}+(-1)^n\}$

$|a_n|=|\frac{1}{n}+(-1)^n|\leq2$ for all integers $n\geq1$. Hence $\{a_n\}$ is bounded.

The first few terms are $\{0,\frac{3}{2},\frac{-2}{3},\frac{5}{4},\frac{-4}{5},\frac{7}{6},\cdots\}$. The subsequence $\{a_{2n}\}=\{\frac{3}{2},\frac{5}{4},\frac{7}{6},\cdots\}$ converges to 1 and the subsequence $\{a_{2n-1}\}=\{0,\frac{-2}{3},\frac{-4}{3},\cdots\}$ converges to -1. Hence $S=\{-1,1\}$

$\liminf a_n=\inf(S)=-1\\ \limsup a_n=\sup(S)=1$