# Conceptual question regarding liminf and limsup for sequences

Let $\{a_j\}$ be a sequence of Real Numbers. Now we define:

$\lim_{j \to \infty}\inf a_j = \lim_{j \to \infty}A_j$

Where $A_j = \inf\{a_j, a_{j+1}, a_{j+2}, \cdots \}$

Now in a line of a proof about a page away, text says to me:

There is a $j_1 \geq 1$ such that $|A_{j_1} - a_{j_1}| < 2^{-1}$

I don't see how this follows from the definition, and so, perhaps, there is something I have missed.

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well, you may recall what is the definition of infimum. From the definition, you can find a $s_k$ as close to the inf as you want, so there should exist the distance between the $A_k$ and $a_k$