Decide which conclusion(s) definitely follow from the chain: Statements: L ≤ M < N > O ≥ P Conclusions: I. O < M II. P ≤ N

Introduction / Context:
We are given a mixed chain and asked which conclusions hold for all assignments consistent with it. We should attempt both proof and counterexample for each conclusion.

Given Data / Assumptions:

• L ≤ M < N
• N > O
• O ≥ P
• Conclusions: (I) O < M, (II) P ≤ N

Concept / Approach:
From O ≥ P and N > O we can relate P and N. For O vs. M, the chain only pins both relative to N, not to each other, so we test by example.

Step-by-Step Solution:
(II) From O ≥ P and N > O, we get N > O ≥ P ⇒ N > P, hence P ≤ N is always true. (I) Comparing O and M: We have M < N and O < N but no direct connector between M and O. O could be less than, equal to (ruled out since O < N and M < N allows equality only if permitted elsewhere), or greater than M without violating the given. For instance, take M = 5, N = 10, O = 9, P = 9 → O > M; or M = 7, N = 10, O = 6, P = 6 → O < M. Since outcomes vary, (I) does not definitely follow.

Verification / Alternative check:
Building explicit numeric models quickly demonstrates (I) is not necessary while (II) is.

Why Other Options Are Wrong:
“Only I”, “either”, and “neither” contradict the deterministic truth of (II).

Common Pitfalls:
Assuming elements on the same “side” of N are ordered with each other, which is not warranted without an explicit relation.

Final Answer:
Only conclusion II follows.