Divergence of Subsequences to Infinity

Prove that if a sequence $(a_n)$ diverges to infinity, then all of its subsequences also diverge to infinity.

When we discuss the divergence of sequences in real analysis, we often focus on how the behavior of a sequence can be characterized when it grows without bound. In this problem, you are tasked with showing that if the original sequence diverges to infinity, then each and every one of its subsequences must also diverge to infinity. This is an important concept as it ensures that the eventual unbounded behavior of the sequence is retained in any of its subsequential paths. The divergence to infinity implies that for every real number, no matter how large, a certain point exists in the sequence beyond which all its terms exceed that number. This condition must naturally carry over to each subsequence since, by definition, a subsequence consists of elements from the original sequence in the same order, albeit potentially less frequently.

Considering subsequences is essential in real analysis because they often reveal intricate properties of sequences that may not be immediately apparent. Specifically, subsequences can help identify the overall trend in a sequence, and understanding their divergence helps solidify our comprehension of the original sequence's behavior. The idea that the entire sequence and all its parts must share this infinite divergence reflects the holistic nature of divergence in real analysis—what is true of the whole is also true of its parts. This exercise not only reinforces your understanding of divergence but also sharpens your ability to rigorously apply definitions and logical reasoning, essential skills for tackling real analysis problems.