Prove there is a limit in a sequence

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SUMMARY

Every monotonically increasing sequence that is bounded from above converges to a limit, as established by the completeness property of the real numbers. This property asserts that every bounded sequence has a least upper bound, or supremum, which serves as the limit of the sequence. The proof can vary depending on the axioms of the real number system being utilized, such as the Archimedean property or the least upper bound property. Engaging with this proof is essential for understanding foundational concepts in real analysis.

PREREQUISITES
  • Understanding of real number axioms, specifically the completeness property.
  • Familiarity with the concepts of monotonic sequences and boundedness.
  • Basic knowledge of limits and convergence in mathematical analysis.
  • Experience with formal proof techniques in mathematics.
NEXT STEPS
  • Study the completeness property of real numbers in detail.
  • Explore the Archimedean property and its implications for sequences.
  • Learn about different methods of proving convergence for sequences.
  • Investigate the role of supremum in real analysis and its applications.
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Mathematics students, educators, and anyone studying real analysis or seeking to deepen their understanding of sequences and limits.

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Prove that every monotonically increasing sequence which is bounded from above has a limit.
 
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Why? This is YOUR homework. If you read the rules for this forum, as you are supposed to have done, then you know you must attempt it and show what you have done. One reason for that (other than the supremely important "you learn by TRYING") is to let us see what you have to work with. I know several different ways to prove this, depending upon which "axioms" you use for the real number system. But I don't know what axioms you are using.
 

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