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: • Definition of the Limi...
Limit Law for Difference of Convergent Sequences: • Proof: Limit Law for D...
Proof that a Constant Sequence Converges to its Constant Value: • Proof: Constant Sequen...
Limit Law for Sum of Convergent Sequences: • Proof: Limit Law for S...
Limit Law for Constant Times a Convergent Sequence: • Proof: Limit Law for C...
Limit Law for Product of Convergent Sequences: • Proof: Limit Law for P...
Limit Law for Quotient of Convergent Sequences: • Proof: Limit Law for Q...
Proving All the Sequence Limit Laws: • Proving All the Sequen...
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[@KhloudAhmed-fc7lj]

how should i come to the assumption and let e = |a|

[@michaelhunt1690]

Hi why did you replace the a(n) by a(N) at 4:36 , wouldn't it have still been correct if we continued with a(n)?

[@guilhermefranco2949]

It suffices to show that some term in the sequence is negative, which contradicts the assumption that they are at least zero for all indices. But if you chose any n > N, it would still work.

[@louiesaunders8853]

Hey sir I have a question for a code question reckon you could help - For the first two digits use an odd number between 39 and 100, for the last two digits use a multiple of 10, repetition is allowed work out the total number of codes which can be made using the method

[@mike_the_tutor1166]

This is not a programming question, it's high-school combinatorics. The word between is ambiguous. Is 39 included? Ask yourself how many odd numbers are between 39 and 100 (including or excluding 39)? Then how many two-digit multiples of 10 are there? Finally, in how many ways could you pair elements of these two sets?