I was focusing on Feller test for explosions for a SDE like this
$$dX_t=\mu(X_t)\cdot dt + \sigma^2(X_t)\cdot dW_t$$
Particularly, I was focusing on Karatzas, Shreve and attention is on exit time of the process $\{X_t\}$ from an interval on the real line $U=(k,l)$. Starting from
$$\tau_{n}=\inf\Big\{t\geq0: \displaystyle{\int}_0^t \sigma^2(X_s)ds\geq n\Big\}; \hspace{0.5cm}n=1,2,...$$
$$T_{u,v}=\inf\{t\geq0: X_t\notin(u,v)\};\hspace{0.5cm}k<u<v<l$$
, after a series of passagges I am not going to mention here, letting both $n$ and $t$ go to $\infty$ I am told that
$$\mathbb{E}\{T_{u,v}\}<\infty$$
that is "$X$ exits from every compact subinterval of $(k,l)$ in finite expected time".
I cannot understand why a conclusion on the exit of the process from every compact subinterval of $(k,l)$ can be drawn. What I mean is:
if I let $t$ go to $\infty$, then $T_{u,v}=\inf\{t\geq0: X_t\notin(u,v)\}=\infty$, which - as far as I know - means that there is not a finite stopping time in which the process $\{X_t\}$ at least touches the bounds of the interval $U$.
However, then the two probabilities
$$\mathbb{P}[X_{T_{u,v}}=u]$$
$$\mathbb{P}[X_{T_{u,v}}=v]$$
are defined and I am told that they add up to $1$, making me guess that either $X_{T_{u,v}}=u$ or $X_{T_{u,v}}=v$, contradicting my above conclusion that the process $X_t$ does never touch (at least) the bounds $u$ and/or $v$.
What's wrong with my reasoning? Could you possibly clear my mind up and/or suggest some other sources which could help me clear my mind up?