STRICT definition of a subsequence.?

Question

I am uncertain whether one is allowed to have repeated elements of a sequence An in a subsequence Bn. If for instance the first few terms of An are:

An = 0, 1, 2, 3, 4, 5, 6, 7...

Is this a possible subsequence:

Bn= 0, 0, 0, 0, 0, 1, 1, 2...

I know elements have to stay ordered, but I am not certain of whether they may be used more than once consecutively. I ask this because as a challenge we were asked to find a sequence from which one could find subseqences that would converge to a) Every rational and b) Every irrational. If repeated elements are allowed then the question could be very simply answered, if not then it is more difficult.

Answer 1

Repeated elements are not permitted in a subsequence, unless of course the elements are repeated in the original sequence. The reason for this is that the index of the subsequence must be "strictly increasing," which means that it cannot stay the same for two consecutive elements.