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Upper and lower bounds are fundamental in mathematics for defining the limits of numerical sets. They are crucial in calculus and algebra for understanding sequences, functions, and set constraints. The least upper bound (LUB) or supremum and the greatest lower bound (GLB) or infimum are the most restrictive limits, essential for discussing convergence and limits. This text delves into their practical calculation and application in various mathematical problems.

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## Definition of Upper and Lower Bounds

### Upper Bounds

Upper bounds indicate the highest value that elements of a set do not exceed

### Lower Bounds

Lower bounds represent the lowest value that elements of a set are not less than

### Importance of Bounds in Mathematics

Upper and lower bounds provide a framework for understanding the constraints and potential extents of numerical sets in various branches of mathematics

## Applications of Upper and Lower Bounds

### Analysis of Sequences, Functions, and Sets

Upper and lower bounds are critical in analyzing the properties of sequences, functions, and sets, acting as numerical constraints that contain the set and facilitate its analysis and comprehension

### Least Upper Bound and Greatest Lower Bound

The least upper bound and greatest lower bound refine the concept of bounds by identifying the most restrictive limits, particularly useful in discussing convergence and limits in calculus

### Impact on Pure Mathematics

Upper and lower bounds play a significant role in pure mathematics, impacting areas such as analysis and topology by defining the extent of sets, functions, and sequences

## Calculating Upper and Lower Bounds

### Determining Bounds

Determining the upper and lower bounds of a set involves examining its elements to ascertain their range

### Strategies for Calculation

Effective calculation of bounds involves recognizing patterns, applying relevant formulas, and using technological tools for visualization and analysis

### Importance of Accuracy

Accurate calculation of bounds is crucial for the integrity of mathematical reasoning and problem-solving methods, requiring careful verification and understanding of the concepts

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