I am working through the way Brian Scott did it in this comment and am wondering why the $-\infty$ case causes a contradiction. https://math.stackexchange.com/a/353697/626797
Question
Let $X_n\leq Y_n$ for each $n\in \Bbb N$. Show that $\liminf X_n \leq \liminf Y_n$ and $\limsup X_n \leq \limsup Y_n$.
Brian Scott response:
Suppose that $X_n\le Y_n$ for each $n\in\Bbb N$, and suppose, to get a contradiction, that $\liminf_nY_n<\liminf_nX_n$. Then there is a subsequence $\langle Y_{n_k}:k\in\Bbb N\rangle$ that converges to some $y<\liminf_nX_n$. Show that either $\langle X_{n_k}:k\in\Bbb N\rangle\to-\infty$, or $\langle X_{n_k}:k\in\Bbb N\rangle$ has a subsequence that converges to some $x\le y$; both are impossible, since $y<\liminf_nX_n$.
I do not see the contradiction during the case when $x\to -\infty$. This implies $y<-\infty$, right? I don't think this is a contradiction and if it isn't, then where is the contradiction during this case?