I've been studying the book Dynamical Systems and Population Persistence, by H. Thieme and H. Smith, and I got stuck in the proof of the first theorem from Chapter 4.
We have a semiflow $\Phi : J \times X \rightarrow X$ on a nonempty set $X$, that is, $J$ is a "time-set": a subset of $\mathbb{R}_{+}$ that contains $0$, $1$ and satisfies $t+s \in J$ whenever $t,s \in J$ and $t-s \in J$ whenever $t,s \in J$ and $t\geq s$, and $\Phi$ satisfies $\Phi(0,x) = x$ for all $x \in X$ and $\Phi(t+s,x) = \Phi(t,\Phi(s,x))$ for all $t,s \in J$ and $x \in X$.
Let $\rho : X \rightarrow \mathbb{R}_{+}$ be the "persistence function". We say that the semiflow is uniformly $\rho$-persistent if there is $\epsilon > 0$ such that $$\liminf_{t \rightarrow \infty} \rho(\Phi(t,x)) > \epsilon$$ whenever $x \in X$ satisfies $\rho(x) > 0$. Accordingly, we also say that the semiflow is uniformly weakly $\rho$-persistent if there is $\epsilon > 0$ such that $$\limsup_{t \rightarrow \infty} \rho(\Phi(t,x)) < \epsilon$$ for all $x \in X$ such that $\rho(x) > 0$. Let $$\sigma(t,x) = \rho(\Phi(t,x)).$$ Then by the semiflow property we have that $$\sigma(t+s,x) = \sigma(t,\Phi(s,x)).$$
We make the following assumptions: There exist a subset $B$ of $X$ and a sequence $(B_k)$ of subsets of $X$ such that the following properties hold:
1) For every $x \in B$, $\sigma(t,x)$ is a continuous function of $t \geq 0$.
2) There are no $y \in B$, $s,t \in J$ such that $\rho(y) > 0$, $\sigma(s,y) = 0$ and $\sigma(s+t,y) > 0$.
3) For every $k \in \mathbb{N}$ and every $x \in X$ such that $\rho(x) > 0$, there exists some $t_k \in J$ such that $\Phi(t,x) \in B_k$ for all $t \geq t_k$, $t \in J$.
4) If $(y_k)$ is a sequence in $X$ with $y_k \in B_k$ for all $k \in \mathbb{N}$, then, after possibly choosing a subsequence, there exists some $y \in B$ such that $\sigma(s,y_k) \rightarrow \sigma(s,y)$ as $k \rightarrow \infty$, uniformly for $s$ in any set $[0,t] \cap J$, $t \in (0,\infty)$.
The Theorem goes on to say that if $J = \mathbb{R}_{+}$ or $J = \mathbb{Z}_{+}$, then under these assumptions the semiflow $\Phi$ is uniformly $\rho$-persistent whenever it is uniformly weakly $\rho$-persistent.
For the proof, we suppose that $\Phi$ is uniformly weakly $\rho$-persistent, but not uniformly $\rho$-persistent. Then there exists $\epsilon > 0$ such that $$\limsup_{t \rightarrow \infty} \sigma(t,x) > \epsilon \mbox{ } \forall x \in X, \rho(x) > 0.$$ On the other hand, let $(\epsilon_j)$ be a sequence in $(0,\epsilon)$ such that $\epsilon_j \rightarrow 0$ as $j \rightarrow \infty$. We can also find a sequence $(x_j)$ in $X$ such that $\rho(x_j) > 0$ and $$\liminf_{t \rightarrow \infty} \sigma(t,x_j) < \epsilon_j.$$
So far, so good. The authors then say that we can find sequences $(r_j),(s_j),(t_j),(u_j),(v_j)$ in $J$ such that $r_j \rightarrow \infty$, $s_j,v_j \leq 1$, and $$\sigma(r_j,x_j) \geq \epsilon, \mbox{ } \Phi(r_j,x_j) \in B_j,$$ $$\sigma(r_j+s_j+t_j,x_j) < \epsilon_j,$$ $$\sigma(r_j+s_j+s,x_j) \leq \epsilon \mbox{ for all } s \in [0,t_j+u_j]\cap J,$$ $$\sigma(r_j+s_j+t_j+u_j+v_j,x_j) \geq \epsilon.$$
I could understand the rest of the proof, but I'm having trouble to find these sequences, especially making them satisfy the second to last condition $$\sigma(r_j+s_j+s,x_j) \leq \epsilon \mbox{ for all } s \in [0,t_j+u_j]\cap J.$$ Any help would be much appreciated. Does anybody know what to do? Thanks!