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.