Proof: Sequence Order Limit Theorem (Inequalities and Limits) | Real Analysis

We prove the order limit theorem for convergent sequences. This proves that convergent sequences and their limits relate to each other and real numbers as we'd expect. For example if a_n converges to a and b_n converges to b, and every a_n is less than or equal to each b_n, then a is less than or equal to b. This theorem summarizes how inequalities behave with limits. #realanalysis

Definition of the Limit of a Sequence: https://www.youtube.com/watch?v=cTnlHZD5ss4
Limit Law for Difference of Convergent Sequences: https://youtu.be/ZlAT0_W4yP0
Proof that a Constant Sequence Converges to its Constant Value: https://www.youtube.com/watch?v=O-quXJi6-ws

Limit Law for Sum of Convergent Sequences: https://youtu.be/Max7ZPWnV1Q
Limit Law for Constant Times a Convergent Sequence: https://youtu.be/W3EWjNSoxqE
Limit Law for Product of Convergent Sequences: https://youtu.be/Zvy860EOlVk
Limit Law for Quotient of Convergent Sequences: https://youtu.be/wBNKbQ-xNiw
Proving All the Sequence Limit Laws: https://youtu.be/dWLhfYUb3wc

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