Prove there is a limit in a sequence

[johnhitsz]

Prove that every monotonically increasing sequence which is bounded from above has a limit.

 

[HallsofIvy]

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.

 

[Architeuthis Dux]

To prove that there is a limit in a sequence, we must show that the terms in the sequence approach a specific value as the index of the sequence approaches infinity.

Let {a_n} be a monotonically increasing sequence that is bounded from above. This means that there exists a real number M such that a_n ≤ M for all n ∈ ℕ.

Since the sequence is monotonically increasing, we know that a_n+1 ≥ a_n for all n ∈ ℕ. This means that the terms in the sequence are getting closer and closer to each other as n increases.

Now, let ε > 0 be a positive real number. We can choose N ∈ ℕ such that a_N > M - ε. This is possible because the sequence is bounded from above.

Since the sequence is monotonically increasing, we know that a_n ≥ a_N for all n ≥ N. This means that for all n ≥ N, we have a_n ≥ M - ε.

Since ε > 0, we can choose a positive integer k such that 1/k < ε. Then, for all n ≥ N, we have a_n ≥ M - 1/k. This means that the terms in the sequence are getting closer and closer to M as n increases.

Therefore, we have shown that for any ε > 0, there exists N ∈ ℕ such that for all n ≥ N, we have |a_n - M| < ε. This is the definition of a limit, and thus we have proved that the sequence {a_n} has a limit of M.

In conclusion, every monotonically increasing sequence that is bounded from above has a limit, which is the supremum of the sequence. This is because the terms in the sequence are getting closer and closer to this supremum as n increases, and we can choose any small margin of error (represented by ε) to show that the terms eventually fall within that margin. This limit is also known as the least upper bound of the sequence.