I'm having trouble with the following statement.
If $M=\liminf_{t\uparrow b}|x(t)|$ then
$\exists$ a sequence $t_n\rightarrow b$ such that
|$x(t_n)| \leq M+1$
I take it that with the sequence $t_n$ we are only interested with the end points. Why aren't all sequences such that $\exists$ an $n_i$ st |$x(t_{n_i})|>M+1$?
I believe that the statement is trying to say is that, since $M$ is the liminf of the function as it approaches $b$, that there must be a sequence $t_n$, which has an image that approaches a number less than M, since you could (in theory) select a sequence which approached $M$. The reason that there does not exist a subsequence for every sequence which has an image that approaches a number greater than $M+1$, is that they make no mention of the limsup here, so the maximum value for the image of a sequence could also be $M$. However, I feel you may have mistyped the > in your question, and meant to write $\leq$, in which case, the reason this isn't true is that the liminf is only known to be true for $t$ which approach $b$, after $b$, the function may very well diverge, we don't know for sure without more information.