Show by induction that (1-nu)(1+u)<=1 for n=0,1,2,3 and u>-1

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In summary, Induction is a mathematical proof technique used to show that a statement or property holds for all natural numbers. It can be used to prove inequalities by first proving the base case and then using the induction step to show that it also holds for the next value of the variable. In this problem, the statement being proved is (1-nu)(1+u) <= 1 for n=0,1,2,3 and u>-1. The restriction u>-1 is necessary to ensure that the expression is defined and to simplify the proof. Induction can only be used for discrete values and within a certain range for n and u.
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gilabert1985
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Hi, I have to solve this problem... I have done something, but I don't know if it is right :/ Thanks a lot for your help!

"Show by induction that (1-nu)(1+u)<=1 for n=0,1,2,3... and u>-1"

For n=0:
(1-0*u)(1+u)^0 <=1
1*1<=1
1<=1, which is true.

Assume that the statement is true for n=k: (1-ku)(1+u)^k<=1

Then it follows that

(1-(k+1)u)(1+u)^(k+1) <= 1... And how do I continue? I really don't have a clue what to do now :(
 
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  • #2
(1-(k+1)u)(1+u)/(1-ku)

Is this expression <=1?
 

Related to Show by induction that (1-nu)(1+u)<=1 for n=0,1,2,3 and u>-1

1. What is induction?

Induction is a mathematical proof technique used to show that a statement or property holds for all natural numbers. It involves proving a base case and then using a series of logical steps to show that the statement holds for the next natural number, thus proving it holds for all natural numbers.

2. How is induction used to prove inequalities?

Induction can be used to prove inequalities by first proving the base case, usually with the lowest value of the variable. Then, assuming the statement holds for some value of the variable, the induction step is used to show that it also holds for the next value of the variable. This can be repeated indefinitely, thus proving the inequality holds for all values of the variable.

3. What is the statement being proved in this problem?

The statement being proved is (1-nu)(1+u) <= 1 for n=0,1,2,3 and u>-1. This means that the expression (1-nu)(1+u) is always less than or equal to 1 for the given values of n and u.

4. Why is the restriction u>-1 necessary?

The restriction u>-1 is necessary because it ensures that the expression (1-nu)(1+u) is always defined and does not result in a division by zero. It also helps to simplify the proof by limiting the range of values that need to be considered.

5. Can induction be used to prove inequalities for any values of n and u?

No, induction can only be used to prove inequalities for discrete values, such as natural numbers. It cannot be used for continuous values, such as real numbers. Additionally, the values of n and u must be within a certain range for induction to be applicable.

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