Sequences and Mathematical Induction by Zhigang Feng

Author:Zhigang Feng
Language: eng
Format: epub
ISBN: 9789811211058
Publisher: World Scientific Publishing Company
Published: 2020-10-15T00:00:00+00:00

Proof. We make use of mathematical induction on n. When n = 1, from the given condition, we know that

So holds true for n = 1.

Now we suppose is true for some n − 1(n 2). We consider n.

Let us denote a = 1 − x1···xn−1, b = yn. From the above equations, we know that in order to prove to be true for n, we only need to prove:

holds true for any a, b ∈ (0, 1). That is to prove

To deal with , we proceed by inducting on m.

When m = 1, obviously is true. Now we suppose to be true for some m − 1 (m 2). Then,

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