Non-Cauchy sequence in $A \times B$

Question

Let $A$ and $B$ be closed subsets of a Banach space $X$. Let $\{(x_n,y_n)\}$ be a sequence in $A \times B$. It is known that if $\{x_n\}$ is not Cauchy then there exists $\epsilon_1>0$ such that for all $k \in \mathbb{N}$ there exists $m_k> n_k \geq k$ for which $\Vert x_{m_k}-x_{n_k} \Vert \geq \epsilon_1$.

Also, if $\{y_n\}$ is not Cauchy then there exists $\epsilon_2>0$ such that for all $t \in \mathbb{N}$ there exists $m_t> n_t \geq t$ for which $\Vert y_{m_t}-y_{n_t} \Vert \geq \epsilon_2$.

Can we say that there exists $\epsilon >0$ such that for all $s \in \mathbb{N}$ there exists $m_s>n_s \geq s$ for which

$$\Vert x_{m_s}-x_{n_s} \Vert \geq \epsilon$$

and

$$\Vert y_{m_s}-y_{n_s} \Vert \geq \epsilon?$$

Answer 1

Yes, take $\epsilon = \min\{\epsilon_1, \epsilon_2\}$.