Divergence of Unbounded Decreasing Sequence

Let $(a_n)$ be a decreasing sequence. Prove that if $(a_n)$ is unbounded, then it diverges to negative infinity.

In this problem, you are asked to prove a fundamental result about sequences: that a decreasing sequence that is unbounded below diverges to negative infinity. This exercise is an exploration of the behavior of sequences in real analysis, where understanding the concept of divergence and how sequences can behave without bound is crucial. The key concept here is the sequence's lack of lower bounds and how its decreasing nature inevitably leads it to negative infinity, illustrating a form of divergence specifically involving negative infinity.

To tackle this problem, remember that a sequence diverging to negative infinity means that for any arbitrarily large negative number, you can find a term of the sequence that is smaller. This requires us to articulate carefully the definitions involved—such as what it means to be unbounded below and to diverge to negative infinity—and to apply these definitions to the decreasing nature of the sequence. We also need to tap into the logical structure of proofs in real analysis, often involving direct proof or proof by contradiction. Pursuing this proof by examining these ideas helps solidify understanding of the behavior of sequences and their limits, particularly how bounds affect their convergence or divergence.