I know the useful fact for sequences stated in this question.
However, suppose I have an arbitrary sequence $\{x_{n}\}$ in $\mathbb{R}$, and I want to prove that $\liminf_{n\to\infty} x_{n} \geq c$ from some $c \in \mathbb{R}$.
We know by definition of $\liminf$ that there exists a subsequence $\{a_{n}\}$ such that: $$\liminf_{n\to\infty} x_{n} = \lim_{n\to\infty} x_{a_{n}}$$ My question is: to complete the proof does it suffice to show that there exists a further subsequence $\{b_{n}\}\subseteq \{a_{n}\}$ such that $\lim_{n\to \infty} x_{b_{n}} \geq c$?
Background: The only reason I feel it is possible is that the first subsequence $\{a_{n}\}$ is, in some sense, the "worst case" subsequence. This makes me feel it might be possible to show that if this worst case subsequence has a subsequence converging to $x \geq c$, then the same might be true for every subsequence of the original sequence. I could then apply the result in the link I posted above.
