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

We use this theorem to prove the functional limit laws: (coming soon)

Check out my real analysis playlists for epsilon-delta limit proofs.

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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[scora]

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!

[이재준]

highschooler here, helped a lot during exams. thanks!

[TukTuk-wf]

Hello, thanks for this proof, I am looking forward to not being forced to use epsilon-delta :D . I came here from your video on function limits - really helpful as well. I'd have a question on the <= direction of this proof: If we assume that the limit does not exits, as you did, would it have been enough to provide one single sequence (if we found one) that defies the generality of our hypothesis as a counterexample? I have something along these lines in my head: "you can use a counterexample to disprove a statement, but it's not enough to prove that it is generally wrong"

[wtt]

Thank you Sir for your very clear explanation ❤

[ShadowWar]

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...

[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.