My problem with the concept of Limit in Real-analysis

[Look]

Hi,

Here is some diagram:

Let the length of the straight orange line be X=1 (X actually can be any finite length > 0).

The rest of the non-straight orange lines (in this particular case, the non-straight orange lines have forms of different degrees of Koch fractal) are actually the same line with length X=1 ,such that its end points are projected upon itself, and as a result we get the convergent series 2*(a+b+c+d+...).

By using the definition of Limit as used by Real-analysis X-2*(a+b+c+d+...)=0, but 2*(a+b+c+d+...) is the projected result of infinite amount of non-straight orange lines, where each one of them has the length X=1.

2*(a+b+c+d+...) (which is the result of the projection of length X=1 upon itself) is actually length 1 only if the projected non-straight line is collapsed into length 0.

This is definitely not the case in the diagram above (there are infinitely many non-straight lines, where each one of them has length X=1).

So, I still do not understand how X-2*(a+b+c+d+...)=0 by using the definition of Limit as used by Real-analysis.

Moreover, if length X=1 is defined in terms of the set of all R members in [0,1], and [0,1] is invariant (exactly as X=1 is invariant in the diagram above), then how exactly |R| is collapsed into cardinality 1 (which is the cardinality of the set with a single member (the vertex at the bottom of the big triangle))?

 

[jvicens]

Hello,

Thank you for sharing your diagram and thoughts on the concept of limit as used in real analysis. It seems like you have a good understanding of the topic and are questioning how it applies to the diagram you have provided.

Firstly, I would like to clarify that the diagram you have shared shows a geometric representation of the concept of limit, rather than a mathematical proof. It is a visual aid to help understand the concept, but it is not meant to be taken as a literal representation of the mathematical equations involved.

In real analysis, the concept of limit is used to describe the behavior of a function as its input approaches a certain value. In your diagram, the input (represented by the length X) is approaching a limit of 1, which is the length of the straight orange line. The non-straight orange lines are used to illustrate how the input is being projected onto itself, resulting in a convergent series.

To address your question about how X-2*(a+b+c+d+...)=0, this equation is not meant to represent the actual lengths of the lines in the diagram. It is simply a mathematical representation of the concept of limit, where the input (X) is approaching the limit (1) and the resulting series (2*(a+b+c+d+...)) is equal to the limit.

As for your question about how the set of all real numbers (|R|) is collapsed into a cardinality of 1, this is not meant to be taken literally. It is a way of representing the idea that as the input approaches the limit, the resulting series converges to a single value.

I hope this helps clarify the concept of limit in real analysis and how it relates to your diagram. If you have any further questions, please let me know. Thank you for sharing your thoughts and engaging in scientific discussion.