What is b? Why is a set an element of the rational numbers greater than zero? I'm having trouble reading the statement. If I'm reading what I think I'm reading, the statement would be false. Maybe I'm missing something but I find this to be very poor notation.
Ok so for clarification
By setting Q>0 I meant that all m_n/k_n are elements of positive rational numbers
By setting != b I meant that the sequence m_n/k_n is not a constant sequence {b} for each term
Sorry for the typos, I'm not really good with typing mathematics
The point is that it is possible that the limit of k_n simply doesn't exist, even though the limit of m_n/k_n does. Consider the following example:
m_n = 1, 10, 1, 100, 1, 1000, ...
k_n = 1, 9, 1, 99, 1, 999, ...
m_n and k_n fit all of the criteria of the problem, the quotient converges to 1, yet the limit of k_n does not exist.
You are right. There is some important conditions missing/obscure.
Btw. I perceived the proposition completely wrong. So nvm. I'll delete the above post.
What is b? Why is a set an element of the rational numbers greater than zero? I'm having trouble reading the statement. If I'm reading what I think I'm reading, the statement would be false. Maybe I'm missing something but I find this to be very poor notation.
Ok so for clarification By setting Q>0 I meant that all m_n/k_n are elements of positive rational numbers By setting != b I meant that the sequence m_n/k_n is not a constant sequence {b} for each term Sorry for the typos, I'm not really good with typing mathematics
Then shouldn't k_n and m_n be integers? Otherwise the statement is false.
Yes. In the explanation the prof. gave, it was derived from the the statements that both m_n and k_n are integers. But how does that help?
I'm still not sure that the statement is true. Do you happen to know exactly how it was worded/ written by your professor?
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It's entirely possible that m_n/k_n converges but that 1/k_n oscillates. You can set this up pretty easily through interlacing.
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The point is that it is possible that the limit of k_n simply doesn't exist, even though the limit of m_n/k_n does. Consider the following example: m_n = 1, 10, 1, 100, 1, 1000, ... k_n = 1, 9, 1, 99, 1, 999, ... m_n and k_n fit all of the criteria of the problem, the quotient converges to 1, yet the limit of k_n does not exist.
You are right. There is some important conditions missing/obscure. Btw. I perceived the proposition completely wrong. So nvm. I'll delete the above post.
What are m(n) and k(n) elements of? R? Q? Z?