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: • Epsilon-Delta Definiti...
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: • Real Analysis
Real Analysis exercises: • Real Analysis Exercises
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.
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[@scora1000]

Hey Wrath, I'm currently taking Fundamental Analysis I. Your videos have helped me a lot in understanding abstract concepts and how to do proofs. They’re clear and concise. Keep up the good work!

[@WrathofMath]

So glad they've helped, thanks for watching! Let me know if you have any questions and good luck in your course!

[@JoeyV115]

I second this

[@wtt274]

Thank you Sir for your very clear explanation ❤

[@phi4321]

Awesome Video! Helped a lot!

[@WrathofMath]

Glad to hear it, thanks for watching!

[@ShadowWar799]

THIS HELPED SO MUCH!! TYSM
can you make a video about protractors

[@SkinnyMMA]

I could not grasp the part where you link the "negated definition of limit of a function " to sequence at 8:12
Would appreciate any help...

[@user-ri9oz3rz1x]

highschooler here, helped a lot during exams. thanks!

[@WrathofMath]

You're very welcome!

[@wannabehuman.production]

I am a bit confused with second direction of the proof. By assuming the limit of f(x) is not L and finding a sequence that defies the rule, isn't that a proof by contrapositive?? I understand the steps in the proof well but i don't understand how it a proof by contradiction.

[@cjjk9142]

its not a proof by contradiction, its just a proof by contrapositive

[@Philologist_as_you_may_see]

It's 12:09am, go sleep bruh

[@like_that4966]

no
who sleeps at 12:09am