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johnhitsz
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Prove that every monotonically increasing sequence which is bounded from above has a limit.
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A limit in a sequence is the value that a sequence of numbers approaches as the number of terms increases. It is the theoretical value that the sequence gets closer and closer to, but may never actually reach.
To prove that there is a limit in a sequence, you must show that as the number of terms in the sequence increases, the terms get closer and closer to a specific value. This can be done using various methods such as the epsilon-delta definition, the squeeze theorem, or the monotone convergence theorem.
Proving a limit in a sequence is important because it allows us to make predictions about the behavior of the sequence as the number of terms increases. It also helps us to determine whether a sequence is convergent or divergent, which has many practical applications in fields such as economics and physics.
No, a sequence can only have one limit. If a sequence has more than one limit, it is considered to be divergent and does not have a limit at all.
Some common mistakes when proving a limit in a sequence include assuming that a specific value is the limit without proper justification, using the wrong definition or theorem, and not considering all possible cases and scenarios. It is important to carefully follow the steps of the proof and provide clear and logical reasoning for each step.