Real analysis - inductive proof- help please

Question

I ran into this problem in my real analysis class. It sounds easy enough on the surface, and maybe it is, but I can’t seem to figure out what to do inductively as the product on the LHS is throwing me a bit off. In short, HELP, please. The question is as follows:

Let n ∈ N. Show that if, a1,a2,...,an are non-negative real numbers less than 1 (0 ≤ ai < 1 for i = 1,...,n), the following holds:

                            1−a1)(1−a2)···(1−an)≥1−(a1 +a2 +...+an)

Can you prove the statement for $n=2$, i.e. with two numbers $a_1,a_2$? — Fawkes4494d3, Sep 03 '20 at 16:22
You can assume that $a_1+\cdots+a_n<1$ since the result is trivial otherwise. — Angina Seng, Sep 03 '20 at 16:28
Please see https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference — Angina Seng, Sep 03 '20 at 16:42

score 0 · Answer 1 · answered Sep 03 '20 at 16:37

0

for n=1 we have (1-a1)>= 1-a1 which is true

now let's assume for n i.e (1-a1)(1-a2).......(1-an)>=1-(a1+........+an) **

and let us show it's true up to n+1

let's multiply ** by (1-an+1) which is positive

(1-a1)(1-a2).......(1-an)(1-an+1)>={1-(a1+........+an){1-an+1}

let's expand the left side we get

1-an+1 -(a1+........+an)+(an+1)(a1+.........+an) simplify

1-(a1+.........+an+an+1)+(an+1)(a1+.........+an)

which is >= 1-(a1+...........+an+an+1)

answered Sep 03 '20 at 16:37

Hasan

57

Please see https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – Angina Seng Sep 03 '20 at 16:42

Real analysis - inductive proof- help please

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