Divergence of a Sequence to Infinity

Consider the sequence $a_n = \frac{n}{2}$. Prove that this sequence diverges to positive infinity.

In this problem, we are tasked with proving that the sequence defined as $a_n = \frac{n}{2}$ diverges to positive infinity. This is a fundamental exercise in understanding the behavior of sequences, particularly those that do not converge to a finite limit. When considering the divergence of sequences, one needs to grasp the idea that a sequence going to infinity means that for every positive number there exists a point in the sequence after which all terms are larger than that number. In simpler terms, no matter how large a number you pick, the sequence will eventually surpass it and continue growing indefinitely.

The problem focuses on the concept of divergence in the realm of real analysis, specifically sequences. The essence of proving divergence to positive infinity involves showing that no upper bound exists for the sequence. This requires a clear definition and understanding of sequences and the arithmetic properties governing them. You'll want to consider how the sequence behaves as n becomes large, and how you can illustrate that beyond any given threshold, the sequence terms exceed it. Grasping this notion equips one with the ability to analyze and predict the behavior of similar sequences and provides a foundation for more complex topics in real analysis.

This type of problem also serves as a bridge to understanding more advanced sequences and series concepts, where convergence and divergence play critical roles. Establishing the divergence of a simple sequence such as this one is a building block for tackling series and the various convergence tests encountered later in real analysis studies. Thus, mastering these skills is crucial for any student delving deeper into mathematical analysis and its applications.