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If we partition a sequence into a finite number of subsequences then the upper and lower limit of the sequence are equal to the maximum upper limit and minimum lower limit of the subsequences.

Has anybody some proof of it, or at least some hint? Or maybe some book that does this proof...

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Here is a hint:

This deals with the $\sup$, the $\inf$ is the same.

The basic idea is that if you have a set $S$ and a partition $P_k$ of $S$ (that is, $S = P_1 \cup \cdots p_n$, the $P_k$ are non-empty and pairwise disjoint), then you have $\sup S = \max_k \sup P_k$.

In the case of the subsequences, each $P_k$ is infinite (corresponding to each subsequence), and the set $S$ is of the form $S = \{n,n+1,... \}$.

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