An Inductive Proof of the Binomial Theorem

In the introductory remedial sections of Kenneth Ross’ Elementary Analysis, the text challenges the reader to prove the oft-utilized binomial theorem. Imagine the omen of my chilling realization of having survived the extent of so-called “engineering math” without having readied familiarity with such a fundamental proof!

Ross specifically calls for the use of an interesting lemma:

Omitting the null case here (for the sake of precious brevity- owed to the mathematical heavens, against which my sins are many), the inductive test:

Having proven the lemma, which proves immensely useful in proving the binomial theorem by allowing the “telescopic-collapsing” of binomial coefficients:

Now, to possess the insight to deconstruct the proof of a larger result into the attainment of intermediate results- not always of the most congruent or relevant appearance, at cursory glance! On-wards and upwards, I suppose…

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