Connecting Function Limits and Sequence Limits | Real Analysis

We prove the limit of a function f as x approaches c is L if and only if the sequence of images of a_n converges to L for all sequences a_n in the domain of f where each a_n is not equal to c. Our bidirectional proof will begin with a direct proof, using the epsilon delta definition of the limit of a function to prove an arbitrary sequence behaves as we desire. Then we use a contradiction proof for the opposite direction, assuming the sequence part of the theorem, but then supposing for contradiction that the limit of the function is not L. #realanalysis Epsilon-Delta Definition of Functional Limits: https://youtu.be/kVQNhAIFZYc We use this theorem to prove the functional limit laws: (coming soon) Check out my real analysis playlists for epsilon-delta limit proofs. Real Analysis Course: https://www.youtube.com/playlist?list=PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli Real Analysis exercises: https://www.youtube.com/playlist?list=PLztBpqftvzxXAN05Gm3iNmpz9SkVfLNqC This theorem on functional limits and sequential limits allows us to use our results about sequences to prove results about functional limits, and also gives us a simple criterion for a function not having a limit: namely, if we find two sequences in the domain converging to c but for which the sequences of their images have different limits, then the function itself does not have a limit at c. ◉Textbooks I Like◉ Graph Theory: https://amzn.to/3JHQtZj Real Analysis: https://amzn.to/3CMdgjI Proofs and Set Theory: https://amzn.to/367VBXP (available for free online) Statistics: https://amzn.to/3tsaEER Abstract Algebra: https://amzn.to/3IjoZaO Discrete Math: https://amzn.to/3qfhoUn Number Theory: https://amzn.to/3JqpOQd ★DONATE★ ◆ Support Wrath of Math on Patreon for early access to new videos and other exclusive benefits: https://www.patreon.com/join/wrathofmathlessons ◆ Donate on PayPal: https://www.paypal.me/wrathofmath Thanks to Petar, dric, Rolf Waefler, Robert Rennie, Barbara Sharrock, Joshua Gray, Karl Kristiansen, Katy, Mohamad Nossier, and Shadow Master 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: https://crayonangel.bandcamp.com/ Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● Facebook: https://www.facebook.com/WrathofMath ● Twitter: https://twitter.com/wrathofmathedu My Math Rap channel: https://www.youtube.com/channel/UCQ2UBhg5nwWCL2aPC7_IpDQ/featured