Suppose {xn} and {yn} are sequences with the following properties:
• xn → L,
• yn → M , and
• xn < yn for all n ∈ N.
TRUE or FALSE: It is necessarily the case that L < M .

The statement is **TRUE**. Here's why:

If we have two sequences \(\{x_n\}\) and \(\{y_n\}\) such that:

1. \(\lim_{{n \to \infty}} x_n = L\),
2. \(\lim_{{n \to \infty}} y_n = M\), and
3. \(x_n < y_n\) for all \( n \in \mathbb{N} \),

then the limit of \(x_n\) (\(L\)) must be **less than or equal** to the limit of \(y_n\) (\(M\)). This is because if \(x_n < y_n\) for all \( n \), as \(n\) becomes arbitrarily large, the values of \(x_n\) and \(y_n\) approach their respective limits while maintaining \(x_n < y_n\).

### Why \(L < M\) is necessary:
If \(L = M\), it would mean that \(x_n\) and \(y_n\) are converging to the same limit, which contradicts the condition that \(x_n < y_n\) for all \( n \in \mathbb{N} \). Therefore, the only possibility is that \(L < M\).

### Conclusion:
Given the conditions, it is necessarily the case that \(L < M\).

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Would you like more details or have any questions? Here are some related questions to consider:

1. What is the relationship between the limits of sequences and their inequalities?
2. What happens if \(x_n \leq y_n\) instead of \(x_n < y_n\)?
3. How would the conclusion change if one or both sequences were not convergent?
4. Can you provide a counterexample if \(x_n\) and \(y_n\) were allowed to have the same limit?
5. How would the result change if \(x_n\) and \(y_n\) were convergent series instead of sequences?

**Tip:** When working with sequences, always consider whether the properties hold in the limit, as small inequalities can become equality as \(n\) becomes large.

Suppose {xn} and {yn} are sequences with the following properties:
• xn → L,
• yn → M, and
• xn < yn for all n ∈ N.
TRUE or FALSE: It is necessarily the case that L < M.

Explore the mathematical proof showing that if two sequences {xn} and {yn} converge to limits L and M respectively, and if xn < yn for all n, then it must follow that L < M. This detailed solution covers key concepts such as the Limit Comparison Theorem and properties of sequences, making it suitable for college-level mathematics students.

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