The discussion revolves around proving that if a sequence {a_n} converges to a non-zero limit A, then the limit of the square root of the sequence, sqrt(a_n), converges to sqrt(A). Participants are exploring the properties of limits and the behavior of sequences under square root transformations.
The conversation includes various attempts to clarify the proof structure and the necessary conditions for convergence. Some participants suggest specific approaches to handle the inequalities, while others express uncertainty about their reasoning or the arrangement of their arguments. There is ongoing exploration of the implications of assuming A > 0 and how to handle cases where A = 0.
Participants note the importance of ensuring that the terms involved in the limits are non-negative, as the square root function is only defined for non-negative numbers in the context of real sequences. There is also a mention of needing to specify the relationship between different epsilon values and their corresponding N values in the proof.
aatzn said:Anyway 1 is not the limit of your sequence, there is no limit for your example.
aatzn said:So, in our case ( \lim a_n = A ), if
\lim \sqrt{a_n}
exists finite, than it is
\sqrt{A}