I am having trouble finding a diverging sequence in R whose subsequences of the even indexed terms.
real analysis converging subsequences
3
$\begingroup$
real-analysis
sequences-and-series
convergence-divergence
divergent-series
2 Answers
1
Let $x_k$ be $k$ if $k$ is prime and $1$ otherwise. Then $\lim_{k \to \infty}x_{nk} = 1$ for all integers $n>1$. However, if $p_k$ is an enumeration of the primes, then we have $\lim_{k \to \infty} x_{p_k} = \infty$.
4
Let the sequence be such that $x_n=0$ if $n$ is a multiple of $2$ or $3$ and $x_n=1$ otherwise.