# Math Help - Convergence

1. ## Convergence

Let $(x_n)$ be a bounded sequence that diverges. Show that there is a pair of convergent subsequences $(x_{nk})$ and $(x_{mk})$ so that

$lim_{k\rightarrow{\infty}}\mid{x_{nk}} - {x_{mk}}\mid$ > 0

2. Originally Posted by yusukered07
Let $(x_n)$ be a bounded sequence that diverges. Show that there is a pair of convergent subsequences $(x_{nk})$ and $(x_{mk})$ so that

$lim_{k\rightarrow{\infty}}\mid{x_{nk}} - {x_{mk}}\mid$ > 0
The equation can equivalently be written $\lim_k x_{n_k}\neq \lim_k x_{m_k}$ since these limits are assumed to exist.

I guess you know that a bounded sequence has a convergent subsequence.

Then choose a convergent subsequence $(x_{n_k})_k$, with some limit $\ell$. Using the fact that initial sequence does not converge, you can find $\epsilon_0$ and a subsequence $(x_{m_k})_k$ such that $|x_{m_k}-\ell|>\epsilon_0$ for all $k$ (I let you justify that); then extract a convergent subsequence from $(x_{m_k})_k$ and conclude.