Proof: Limit Law for Constant Times a Convergent Sequence | Real Analysis

We prove the limit law for a constant multiplied by a convergent sequence. If a_n converges to a and c is a real number, then the sequence c*a_n converges to c*a. As in, the constant multiple of a convergent sequence converges to its limit multiplied by the same constant. This is a straightforward proof using the epsilon definition of a convergent sequence, except for the case where c is 0. If c = 0, our result follows from the fact that a convergent sequence converges to its constant value.

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
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 Difference of Convergent Sequences: https://youtu.be/ZlAT0_W4yP0
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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