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I know $n \in \mathbb{N}$ and...

$$ a_n = \begin{cases} 0 & \text{ if } n = 0 \\ a_{n-1}^{2} + \frac{1}{4} & \text{ if } n > 0 \end{cases} $$

  1. Base Case:

$$a_1 = a^2_0 + \frac{1}{4}$$

$$a_1 = 0^2 + \frac{1}{4} = \frac{1}{4}$$

Thus, we have that $0 < a_1 < 1$. So our base case is ok.

  1. Inductive hypothesis:

Assume $n$ is arbitrary. Suppose $$0 < a_{n} < 1$$ $$0 < a_{n-1}^{2} + \frac{1}{4} < 1$$ is true, when $n > 1$.

  1. Inductive step:

Let's prove $$0 < a_{n+1} < 1$$ $$0 < a_{n}^{2} + \frac{1}{4} < 1$$

is also true when $n > 1$.

My guess is that we have to prove that $a^2_{n}$ has to be less than $\frac{3}{4}$, which otherwise would make $a_{n+1}$ equal or greater than $1$.

So we have $(a_{n-1}^{2} + \frac{1}{4})^2 < \frac{3}{4}$... I don't know if this is correct, and how to continue...

  • It isn't. You prove inductively that $0 \leqslant a_n \leqslant \frac{1}{2}$ for all $n$. – Daniel Fischer Nov 21 '14 at 15:47
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    What he's trying to say is: Pretend that the problem was "Prove that $0\leq a_n \leq \frac{1}{2}$". – Arthur Nov 21 '14 at 15:54
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    No. When you are inductively proving $0 \leqslant a_n \leqslant \frac{1}{2}$, in the induction step, you have the stronger assumption $0 \leqslant a_n \leqslant \frac{1}{2}$, which suffices to conclude $0 \leqslant a_{n+1} \leqslant \frac{1}{2}$. – Daniel Fischer Nov 21 '14 at 15:54
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    @nbro what Daniel is telling you to do is to prove something that implies what you want to prove. Instead of proving by induction that 0< $a_n$ < 1, you prove by induction that 0 < $a_n$ < $\frac{1}{2}$. And the latter implies that 0 < $a_n$ < 1 – mvggz Nov 21 '14 at 15:55
  • @nbro you mean 0 < $a_n$ < $\frac{1}{2}$ ? It's simpler than what you think :) . You want to prove the property P : " 0 < $a_n$ < 1 ". You have several tools for this and among them, induction. But you can use something else. You can find a property A such as : A => P while being easier to prove by induction than P. So you will use induction to prove A, and then you'll know that A=> P, so you will have proven P but you'll have used induction on A :) – mvggz Nov 21 '14 at 16:02
  • A:" 0 < $a_n$ < $\frac{1}{2}$ " ; P:" 0 < $a_n$ < 1 " We agree on the fact that : A => P Use induction to prove A: 0 < $a_n$ < $\frac{1}{2}$ => 0 < $a_{n+1} = a_n^2 + \frac{1}{4} $ < $\frac{1}{4} + \frac{1}{4} = \frac{1}{2}$ A has been proven by induction. Now since A => P, you have proven P to be right by proving A using induction, clearer? – mvggz Nov 21 '14 at 16:08
  • @nbro you don't try to solve an inequality, you assume an inequality is true for $a_n$ and using it you want to prove that this very same inequality is also true for $a_{n+1}$ – mvggz Nov 21 '14 at 16:10
  • @nbro I'll add a more thorough answer, it's ok :) . Everybody has the right to be disoriented by something :), has nothing to do with stupidity Tell me if the answer is clear enough or not, I can edit – mvggz Nov 21 '14 at 16:18

3 Answers3

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Starting with $a_0 = 0$, it is easier to show the stronger inequality, $0 < a_n < 1/2$ for $n \in \mathbb{Z}^+$. This conclusion immediately falls out from the recurrence relation.

Adhvaitha
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Ok I'll right it down so that it's clear to you :)

I want to prove the property P: " 0 < $a_n$ < 1 "

I look at the property A: $0 < a_n < \frac{1}{2}$

A => P, that is : If A is true then P is true

I'll prove A using induction (so technically I don't prove P by induction, but by implication).

$ a_o = 0 $ < $\frac{1}{2}$

If $ 0 < a_n < \frac{1}{2} $ , then :

$ a_{n+1} = a_n^2 + \frac{1}{4} $ > $a_n^2 $ > 0

And : $ a_{n+1} = a_n^2 + \frac{1}{4} $ < $ (\frac{1}{2})^2 + \frac{1}{4} = \frac{1}{2} $

Hence you get : 0 < $a_{n+1}$ < $\frac{1}{2}$ : the hypothesis holds for the rank n+1

So you have proven using induction that for every n positive integer you have :

0 < $a_n$ < $\frac{1}{2}$

But since : $\frac{1}{2}$ < 1 , you also have:

0 < $a_n$ < $\frac{1}{2}$ < 1 ie 0 < $a_n$ < 1

mvggz
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  • @nbro Of course I do, that's what the induction step is. The principle of induction is that you verify the property for the first terms , and then you suppose it is true for a arbitrary rank , usually called "n". Then, if you can prove that the hypothesis at the rank "n" implies that it is true for the rank n+1, you can conclude that it is true for every n integer. So I have to start the induction by saying "if the property 0 < $a_n$ < $\frac{1}{2}$ is true, then ..." and I finally conclude by something like " then.. 0< $a_{n+1}$ < $\frac{1}{2}$ . That's the process, you've got to grasp this – mvggz Nov 21 '14 at 20:04
  • I've edited my answer, tell me if it makes more sense to you, but I can assure you that my reasoning is correct :) – mvggz Nov 21 '14 at 20:10
  • @nbro, that's something that needs practice for sure :) , you'll get used to it and even have an intuitive perception throughout the learning – mvggz Nov 21 '14 at 20:15
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Note: Base on the mvggz's answer, I will try to give also my complete with a lot of explanations answer. If something is wrong, please write in the comment section below ;)

Let $$ a_n = \begin{cases} 0 & \text{ if } n = 0 \\ a_{n-1}^{2} + \frac{1}{4} & \text{ if } n > 0 \end{cases} $$

For all $n \in \mathbb{N}$ and $n \geq 1$

  1. Basis:

Our base case is when $n = 1$, so let's verify the following statement $$0 < a_1 < 1$$ We know that $a_1 = a^2_0 + \frac{1}{4}$, so we have: $$0 < a^2_0 + \frac{1}{4} < 1$$ $a_0$ is 0, from the definition of $a_n$, so we have: $$0 < 0^2 + \frac{1}{4} < 1$$ $$0 < \frac{1}{4} < 1$$ which is clearly true, so our base case is proved.

  1. Inductive step

Let's assume that $0 < a_n < \frac{1}{2}$ (this is a stronger case, because we are assuming that $a_n < \frac{1}{2}$ instead of $<1$)

We want to prove that $0 < a_{n+1} < 1$, but if we prove $0 < a_{n+1} < \frac{1}{2}$, then we prove also the first one, because $\frac{1}{2} < 1$.

We know that:

$$a_{n+1} = a^2_n + \frac{1}{4} > a^2_n > 0$$

We know $a^2_n > 0$, because $a_n$ is a positive number, and even though it wasn't, it would become positive, because raised to the power of $2$, would make it positive.

Since we assume that $a_n < \frac{1}{2}$, suppose we replace $a_n$ with $\frac{1}{2}$, and we say that (note the $<$ sign):

$$a_{n+1} = a^2_n + \frac{1}{4} < \left( \frac{1}{2} \right)^2 + \frac{1}{4}$$ $$a_{n+1} = a^2_n + \frac{1}{4} < \frac{1}{4}+ \frac{1}{4}$$ which simplified becomes: $$a_{n+1} = a^2_n + \frac{1}{4} < \frac{1}{2}$$

and we have just proved that $a_{n + 1} < \frac{1}{2}$, so it must also be less than $1$. Note that $a_{n+1} > 0$, because $a^2_n > 0$ and also $\frac{1}{4} > 0$.