Hello, I had this on my exam today, I was positive that I'd be able to deal with inequalities but for some reason this one threw me off. I had to prove

for all positive integers i.e n=0,1,2,3..

anyone mind throwing some hints?

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- Mar 16th 2010, 10:51 PMDave2718Mathematical Induction with an inequality
Hello, I had this on my exam today, I was positive that I'd be able to deal with inequalities but for some reason this one threw me off. I had to prove

for all positive integers i.e n=0,1,2,3..

anyone mind throwing some hints? - Mar 17th 2010, 02:50 AMemakarov
This holds iff . To prove this by induction, note that when going from to , the left-hand side is multiplied by , while the right-hand side is multiplied by .

In fact, it is not necessary to take reciprocals. The LHS is divided by , and the RHS is divided by 2. - Mar 17th 2010, 04:29 AMArchie Meade
- Mar 17th 2010, 07:35 AMnovice
It's straight forward.

I will show you the inductive step.

*Proof:*

We assume , and wish to prove .

By implication it is expressed as

So if we can start from and end our proof with , we are home free.

Let's begin with . Multiplying through with , we obtain

. Observe

So we succeeded. - Mar 17th 2010, 11:40 AMArchie Meade
- Mar 17th 2010, 12:13 PMnovice
- Mar 17th 2010, 02:12 PMDave2718
thanks guys, I actually tried the problem after the test before anyone was able to help me out and actually got it pretty much just as all of you did, don't you hate it when that happens?