Proof of the Multiplicativity of Sequence Convergence
TYPO ALERT: At the 7:27 mark, I say we are using the triangle inequality but write an equality. It should be less than or equal to, not equal to.
We prove that if (a) converges to K, and (b) converges to L, then the product sequence (ab) converges to KL. The proof proceeds similar to the proof of the additivity of sequence convergence, only there is a tricky step (adding and subtracting the same quantity). We also use the triangle inequality, the inspired choice theorem, as well as the result from a previous video that shows that a convergent sequence is bounded from above and below.
Inspired Choice Theorem: • Proof of the Inspired Choice Theorem
Convergent Sequences are Bounded: • Proof that Convergent Sequences are B...
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