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 Definition of Functiona...  
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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