Kalman Noel wrote:
Ben Franksen wrote:
Kalman Noel wrote:
(2) lim a_n = ∞
[...]
(2) means that the sequence does not converge, because you can always find a value that is /larger/ than what you hoped might be the limit.
(2) usually rather mean that for each positive limit A there is a number N such that a_N > A for /all/ n > N.
You're right here. I tried to come up with a more wordy, informal description, but failed on that.
Your definition of (2) is usually termed as '(a_n) contains a subsequence that tends toward +infinity'.
May you elaborate? I don't see where a subsequence comes into play here.
I'll show (2) <=> (2'), where (2'): (a_n) contains a subsequence that tends toward +infinity "=>" : Assume (2) holds. Construct a subsequence (b_m) of (a_n) by chosing, for each natural number m, an index n_m such that b_m = n_(n_m) is larger than m (which is possible by (2)). Then (b_m) is a subsequence of (a_n) that tends toward infinity (as I defined it). "<=" : Assume (2') holds. Let A > 0 be any positive number (that you "hope might be the limit"). We want to show that we can find N such that a_N > A. To do so, chose a M, such that b_M > A (which is possible by assumption). Then there exists an N such that a_N = b_M, because (b_n) is a subsequence of (a_n). q.e.d Cheers Ben