Why are these sufficient conditions for $\liminf$?

Question

My professor today stated that to show that $\liminf a_n=A$ it is sufficient to show

If

$1.\,a_n\geq b_n,\,\,\forall n\in\mathbb{N}$

$2.\,a_{n_k}\leq c_k,\,\,\forall k\in\mathbb{N}$

$3.\lim c_k=\lim b_n=A,\,\,n,k\to\infty$

then

$\liminf a_n=A,\,\,n\to\infty$

What is the idea or intuition behind this? I somewhat see a squeeze theorem and maybe(?) the use of Cauchy sequences, but it is not clear to me how we can go from needing to evaluate $\liminf$ to simply evaluating $\lim$.

Answer 1

Let $D = \liminf a_n$. We have that $D \leq \lim \inf a_{n_k}$, since the $\lim \inf$ of the full sequence is at most the $\liminf$ of any subsequence. Since $\lim \inf a_{n_k} \leq \lim \inf c_k = \lim c_k = A$, we get that $D \leq A$.

On the other hand, since $a_n \geq b_n$ for each $n$, $D \geq \lim \inf b_n = \lim b_n = A$.

Thus $D=A$.