Proof: Limit Law for Sum of Convergent Sequences | Real Analysis

We prove the limit law for the sum 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 sum of convergent sequences converges to the sum of their limits. This is a fairly straightforward proof using the epsilon definition of a convergent sequence.
Proof of Triangle Inequality Theorem: • Proof: Triangle Inequa...
Proof of Reverse Triangle Inequality: • Proof: Reverse Triangl...
Definition of the Limit of a Sequence: • Definition of the Limi...
What are Bounded Sequences? • What are Bounded Seque...
Proof that a Convergent Sequence is Bounded: • Proof: Convergent Sequ...
Limit Law for Difference of Convergent Sequences: • Proof: Limit Law for D...
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...
Real Analysis Playlist: • Real Analysis
★DONATE★
◆ Support Wrath of Math on Patreon for early access to new videos and other exclusive benefits: / wrathofmathlessons
◆ Donate on PayPal: www.paypal.me/...
Thanks to Robert Rennie and Barbara Sharrock for their generous support on Patreon!
Thanks to Crayon Angel, my favorite musician in the world, who upon my request gave me permission to use his music in my math lessons: crayonangel.ba...
Follow Wrath of Math on...
● Instagram: / wrathofmathedu
● Facebook: / wrathofmath
● Twitter: / wrathofmathedu
My Music Channel: / @emery3050

[@jingyiwang5113]

You have offered a really nice proof. Thank you so much!

[@WrathofMath]

Glad to help! Thanks for watching and check out my analysis playlist if you're looking for more! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@joshtumacay6497]

Hello, just wanna ask, in 5:03 how did you arrive at ε/2? Thanks :)

[@christian.adriano]

Great videos! Suggestion - Maybe if you add a number prefix in the title of the video, it would make it easier to follow from one to another in an optimal sequence and also help categorize them in sub-topics.

[@laith4319]

Excellent video! Other proofs for this concept online (and in my my notes) did not go into nearly as much detail behind the rationale for the steps👍

[@MUSHIN_888]

My lecturer just passed through this and only after coming back to revise this I realised how much information and connotations he left out.

[@martin_skachkov6016]

Hey, great video! Are there videos for the multiplication and division like this one?

[@Leyla-et9nh]

İs the converse also True?

[@WrathofMath]

Thanks for watching and no it is not. To see why, imagine { a_n + b_n } converges to 1. So then what? Does a_n converge to 10 with b_n converging to -9? Or does a_n converge to pi with b_n converging to 1-pi? Or...you see the issue. It could very well be the case that a_n and b_n both actually diverge, but have a sum converging to some finite value.

[@Leyla-et9nh]

@@WrathofMath thank you very much 💕

[@QuantitativeDiaries]

Love u

[@WrathofMath]

love u 2