We prove the limit law for the product of convergent sequences. If a_n converges to a and b_n converges to b, then the sequence a_n*b_n converges to a*b. As in, the product of convergent sequences converges to the product of their limits. This is a slightly tricky proof using the epsilon definition of a convergent sequence, and absolute value manipulations!
Proof of Triangle Inequality Theorem: https://www.youtube.com/watch?v=051WWg9twO0
Proof of Reverse Triangle Inequality: https://www.youtube.com/watch?v=4OmIRB5Zv-4
Definition of the Limit of a Sequence: https://www.youtube.com/watch?v=cTnlHZD5ss4
What are Bounded Sequences? https://www.youtube.com/watch?v=hkCKQdAFLAw
Proof that a Convergent Sequence is Bounded: https://www.youtube.com/watch?v=7N-4XZlxBho
Limit Law for Difference of Convergent Sequences: https://youtu.be/ZlAT0_W4yP0
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
Real Analysis Playlist: https://www.youtube.com/playlist?list=PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli
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