MHB What is the mistake in this supposed proof?

[evinda]

Hello! (Wave)

We consider the following problem

$$y'(x)=-sign y, y(0)=0$$

where $sign y$ is defined as follows:

$$sign y=\left\{\begin{matrix}
1 & , y \geq 0\\
-1 &,y<0
\end{matrix}\right.$$The consecutive approaches are:$$\phi_m(x)=\left\{\begin{matrix}
-x &, m=1,3,5,7, \dots \\
|x| &, m=2,4,6,8,\dots
\end{matrix}\right.$$

Find the mistake at the following syllogism:We consider the subsequence $\phi_k(x), k=2m+1, m=0,1,2,3, \dots$ and we take the limit , obviously

$$\lim_{k \to +\infty} \phi_k(x)=\lim_{k \to +\infty} (-x)=-x=\phi(x)$$Consequently $\phi(x)=-x$ is the solution of the problem in the whole $\mathbb{R}$.We have $\phi_k(x)=y_0+\int_{x_0}^x f(\xi, \phi_{k-1}(\xi)) d \xi$Can we not take the limit since $\phi_k$ and $\phi_{k-1}$ don't have the same value?In my notes it stands the following:$$1,2, \frac{1}{2}, 2, \frac{1}{3},2, \frac{1}{4},2, \dots \\ \frac{1}{n} \to 0$$

How do we find these values?

 

[Evgeny.Makarov]

Where exactly is the syllogism?

 

[evinda]

Where exactly is the syllogism?

This part:

We consider the subsequence $\phi_k(x), k=2m+1, m=0,1,2,3, \dots$ and we take the limit , obviously

$$\lim_{k \to +\infty} \phi_k(x)=\lim_{k \to +\infty} (-x)=-x=\phi(x)$$Consequently $\phi(x)=-x$ is the solution of the problem in the whole $\mathbb{R}$.

 

[Evgeny.Makarov]

A syllogism is a technical term in logic, and its meaning is an inference rule or its application. There are many different types of syllogisms. Some of the most famous are:

(1) $\forall x\in A\,P(x)$; $x_0\in A$; therefore, $P(x_0)$.

(2) $P$ implies $Q$; $Q$ implies $R$; therefore, $P$ implies $R$.

I don't see how your reasoning fits this pattern precisely. In particular, it is not clear what theorem you are using to conclude that $\phi(x)=−x$ is the solution of the problem.

 

[evinda]

A syllogism is a technical term in logic, and its meaning is an inference rule or its application. There are many different types of syllogisms. Some of the most famous are:

(1) $\forall x\in A\,P(x)$; $x_0\in A$; therefore, $P(x_0)$.

(2) $P$ implies $Q$; $Q$ implies $R$; therefore, $P$ implies $R$.

I don't see how your reasoning fits this pattern precisely. In particular, it is not clear what theorem you are using to conclude that $\phi(x)=−x$ is the solution of the problem.

I didn't write it by myself , it is given at an exercise and I should deduce if it is right or wrong and justify why it is like that.

 

[Evgeny.Makarov]

Well, first you may let your teacher know that "syllogism" is not the best term here, in my opinion. Perhaps he or she meant "reasoning", "conclusion", or "supposed proof".

Second, problems of this sort usually invoke some theorem, whose conclusion is paradoxical in that particular case. This usually happens when one of the assumptions of the theorem is not satisfied. But it should be clear which theorem is used. You should know better what topic your course is covering now. What are the approximations, where are they coming from?