Suppose we have a sequence $\{a_1,...,a_n,...\}$ with $\underset{n\rightarrow\infty}{\liminf} a_n=A$. How to find the smallest value in this sequence? My current intuition is that the smallest value might not always be attained in this sequence. And the only case in which the smallest value is obtained is when there exists relatively small values in the beginning of the sequence. In this case, for some given small $\epsilon>0$, I can always find $N$ such that the subsequence of small numbers denoted by ${a_{k_n}}$ with $k_n>N$ will fall within $[A-\epsilon,A+\epsilon]$. With relatively small values in the beginning of the sequence, I can find a number among $a_1,..,a_N$ that is smaller than $A-\epsilon$ which will be the smallest value in this sequence.
Is my intuition correct?