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The concept of density at a point is essential for understanding how a set behaves around a particular location. Specifically, density at a point within a set helps determine the proportion of the set that exists near that point. Let’s consider a point, say 0, within the real number line \(\mathbf{R}\). To calculate the density of a set \(A\) at the point 0, we use the formula:

\[\lim_{r \to 0}\frac{\text{Length}(A \cap (-r, r))}{2r}.\]This formula essentially examines how much of set \(A\) lies within an interval around the point 0 as the interval size \(r\) shrinks closer to zero. By understanding this density, we can ascertain how densely packed the set is around that particular point. For example, if this limit value is \(\frac{1}{3}\), it means near the point 0, the set \(A\) takes up one-third of the interval around 0, providing a quantitative measure of distribution.

Nested intervals refer to a sequence of decreasing intervals that are contained one within another. This process is significant in many mathematical constructions and proofs. In our step by step example, the nested intervals start with a larger interval and progressively shrink in size. The intervals are:

- \((-1, 1)\) with length 2,- \((-1/3, 1/3)\) with length 2/3,- \((-1/9, 1/9)\) with length 2/9, and so forth.

This structured pattern helps set the stage for creating sequences with contained properties. The concept is essential when constructing sets like Borel sets, where the intersection or union of these nested intervals possess certain characteristics desired in the final set. Nested intervals ensure that each subsequent interval is always "nested" within its predecessor, helping achieve precision in mathematical constructions.

In mathematics, the limit of a sequence is a fundamental concept used to describe the value that elements of a sequence tend towards as the index goes to infinity. For instance, when constructing the set \(A\) as explained, we analyze the convergence of the lengths of selected intervals as each interval \(I_k\) contributes a part \(J_k\) to the construction.

The key step involves evaluation of an infinite sum limit \[\lim_{r \to 0} \frac{1}{r} \sum_{k: 1/3^k < r} \frac{1}{3^{k+1}}.\]This represents how closely the set \(A\) adheres to occupying \(\frac{1}{3}\) of any small interval around 0 as \(r\) approaches zero. Understanding such limits allows mathematicians to define precise behaviors of sequences and functions in analysis.

The construction of mathematical sets, such as Borel subsets, often involves careful consideration of properties that the set must exhibit. In the given task, the goal was to construct a Borel subset of \(\mathbf{R}\) with a specified density at a single point.

To achieve this:

• Create a sequence of nested intervals centered at the target point.
• Within each interval, select portions that fit the density condition.
• Use union of these portions to form the final set \(A\).

The resulting set \(A\) must be expressed as a Borel set to align with the desired properties of measurability. By carefully constructing \(A\) through this method, we ensure that its behavior coincides with the required density near the designated point, demonstrating the application of theoretical principles in real, tangible examples.