Unbounded Increasing Sequence Diverges to Infinity

Let $(a_n)$ be an increasing sequence. Prove that if $(a_n)$ is unbounded, then it diverges to positive infinity.

This problem delves into the fundamental concepts of sequences within real analysis, particularly focusing on the behavior of increasing sequences. An increasing sequence is one where each term is larger than or equal to the previous term. The key aspect to consider here is the notion of boundedness. A sequence being unbounded means that there is no real number that serves as an upper limit for all terms of the sequence. In other words, given any real number, no matter how large, there exists a term in the sequence greater than that number.

When dealing with unbounded increasing sequences, an important characteristic is their divergence to positive infinity. Divergence, in this context, implies that the terms of the sequence grow without bound. Proving this involves showing that for any positive number, there exists a term in the sequence beyond which all subsequent terms exceed that number. This property captures the essence of divergence to infinity.

In tackling such problems, it's crucial to leverage the definition of sequence convergence and relate it to the contrapositive argument: if a sequence does not converge to a finite limit, and yet is increasing, it must diverge to positive infinity. Understanding these relationships builds a robust foundation for further exploration of more intricate concepts in sequences and convergence within real analysis.