Prove there is a limit in a sequence

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Every monotonically increasing sequence that is bounded from above converges to a limit due to the completeness property of the real numbers. This property states that every non-empty set of real numbers that is bounded above has a least upper bound, or supremum. The discussion emphasizes the importance of attempting the proof independently to facilitate learning and understanding. Different axioms of the real number system can lead to various proof methods, highlighting the need for clarity on which axioms are being utilized. Engaging with the problem directly is crucial for grasping the underlying concepts.
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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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