Order Limit Theorem for Convergent Sequences

Prove the order limit theorem for convergent sequences, including the three statements:

1) If every term of a convergent sequence is at least 0, then the limit of that sequence is also at least 0.

2) If every term of one convergent sequence is less than or equal to some other convergent sequence, then their limits have that same relation.

3) If a convergent sequence is bounded below by some real number, then the limit of that convergent sequence is at least as big as that lower bound. Similarly, if a convergent sequence is bounded above by some real number, then its limit is less than or equal to that real number.

The Order Limit Theorem is a cornerstone in the study of sequences and their behavior as they converge toward limits. Understanding this theorem involves grappling with the intrinsic properties of real numbers and sequences, focusing especially on the preservation of order through the limiting process.

The theorem fundamentally asserts how order relations are maintained when sequences converge, underscoring three key propositions. Firstly, a sequence where each term is non-negative will result in a non-negative limit. This illustrates the stability of the non-negative property, even as the sequence advances toward infinity.

Secondly, if each term of one convergent sequence is bounded above by terms of another, the limits will uphold this order, reflecting the sequences' consistent upper and lower constraints in their behavior towards convergence.

Lastly, the theorem addresses bounded sequences, maintaining that if a sequence is constrained within set bounds, the limit will respect those bounds. These insights are crucial as they form a bridge from sequences to more complex concepts, helping to understand how individual behaviors within sequences predict their convergence characteristics.

These propositions reveal the predictability of limits based on sequence characteristics, offering a foundational understanding that supports further studies in analysis, such as series convergence and the intricacies of function behavior.