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Convexity is a fundamental concept in mathematics, particularly useful in areas like optimization, economics, and geometry. In simple terms, a set is convex if, for any two points within the set, the line segment that joins them is also entirely contained within the set.

For real numbers, this definition translates into a visually intuitive idea: imagine plotting points on a number line. If for any two points you pick within a set you can sweep a pencil in a straight line from one to the other without leaving the set, then that set is convex. If you find yourself lifting the pencil because there's a 'gap', the set is not convex.

One of the most common examples of convex sets in real numbers are intervals. Intervals can be open (not including endpoints), closed (including endpoints), or semi-open (including one endpoint but not the other). The key characteristic of an interval is that it contains all the numbers between its endpoints, fitting perfectly within the definition of a convex set.

In the context of the real numbers, an interval is a set of numbers lying between two specific numbers, known as the endpoints. The beauty of intervals is in their simplicity and the ease with which we can describe them using inequality notation or interval notation.

There are several types of intervals, and they can be categorized into closed intervals \[a, b\], which include the endpoints a and b, open intervals \(a, b\), which exclude the endpoints, and half-open (or half-closed) intervals \[a, b\) or \(a, b\], which include only one of the endpoints.

Understanding intervals is essential since many mathematical concepts rely on this foundational idea. Real-life applications often require us to work with ranges of values 鈥� whether we're considering the optimal temperature range for a chemical reaction, defining the span of years for historical research, or setting thresholds for financial investments.

Proofs are the backbone of mathematical truth. They are rigorous arguments that are constructed to demonstrate the truth of a proposition or theorem. Various proof techniques are used to establish the veracity of theorems within mathematics. Some of these techniques include direct proof, proof by contradiction, proof by induction, and proof by contraposition.

In the case of proving statements about convex sets and intervals, direct proofs are often employed. A direct proof typically starts with a hypothesis and uses the definitions and properties that come from our understanding of mathematics to arrive at a conclusion. The steps are logical and straightforward, intended to show that if the initial hypothesis is true, then the resulting conclusion must also be true.

For the exercise in question, direct proof is appropriate because the relationship between convexity and intervals can be shown by using their definitions. The proof essentially says, 'Given the definition of convexity in real numbers and what we know about intervals, we can prove conclusively that a subset of real numbers is convex if and only if it is an interval.'