Sequence with set of limit points as all rational numbers

To construct a sequence whose set of limit points is only the set of rational numbers, we can follow the steps below:

Step 1: Enumerate the Rational Numbers
First, we need to enumerate the set of rational numbers. We can do this by arranging the rational numbers in a table or grid pattern. For example, we can start with the following enumeration:

1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
...

Step 2: Traverse the Rational Numbers
Next, we need to traverse the enumeration of rational numbers in a specific order to construct our sequence. One possible order is to start with the first row, then move diagonally to the second row, then diagonally to the third row, and so on. Within each row, we can traverse from left to right.

Following this order, we can construct the sequence as follows:

1/1, 1/2, 2/1, 3/1, 2/2, 1/3, 1/4, 2/3, 3/2, 4/1, ...

Step 3: Analysis of the Sequence
Now, let's analyze the properties of this sequence.

Every Rational Number Appears in the Sequence
Since we are traversing through the enumeration of rational numbers, every rational number will eventually appear in the sequence. Therefore, the set of limit points of this sequence includes all rational numbers.

No Irrational Numbers as Limit Points
On the other hand, this sequence does not contain any irrational numbers as limit points. This is because the sequence is constructed by traversing through a grid pattern of rational numbers, and irrational numbers are not included in this pattern.

Conclusion
In conclusion, the sequence constructed by traversing through the enumeration of rational numbers in a specific order has the set of limit points as all rational numbers. This sequence provides an example of a sequence whose set of limit points consists only of rational numbers and does not include any irrational numbers.