In other words: if [alpha] < [OMEGA] is of second kind (ordinal limit), then there exists a strictly ascending sequence {[[alpha].sub.n]} of countable ordinals with property (3).

Assume that [beta] is an ordinal limit; that is, there exists a strictly ascending sequence [([[beta].sub.n]).sub.n] of ordinals in [OMEGA] such that [beta] = sup{[[beta].sub.n]; n [member of] N} and [[beta].sub.m] [??] [[beta].sub.n] < [beta] whenever m [less than or equal to] n.

A tweet ID is an integer ID attached to all tweets in ascending sequence since the Twitter service began.

Procedure A: Acquiring a tweet ID of almost the last tweet of the next day of the target date Input: target date [d.sub.q], target cell of a grid [c.sub.q] Output: the tweet ID of almost the last tweet of the next day of the target date [id.sub.q] access Twitter Search API specifying date as the next day of [d.sub.q] and area as [c.sub.q], and acquire a list of tweets T in ascending sequence of tweet IDs (100 tweets are acquired according to the specification of the API) set [id.sub.max] to the tweet ID of the last item in T (i.e.

(b04) each ascending sequence in Dom([phi]) has an upper bound (in M).

We have thus constructed an ascending sequence ([u.sub.n]) in [M.sub.u] [[subset].bar] Dom([phi]) for which ([phi]([u.sub.n])) is (strictly) descending and bounded below; hence [lambda] := [lim.sub.n] [phi](un) exists in R.

Art Connoisseur seems to be progressing in a similar fashion to his sire Lucky Story, who produced an

ascending sequence of Racing Post Ratings on his five starts at two, culminating in a 117 in the Group 2 Champagne Stakes, and it is possible that Art Connoisseur, who has shown marked improvement on each of his three starts to post RPRs of 83, 99 and 113, will be able to challenge the 120 RPR barrier next time we see him.

In a seminal 1943 paper, he divided human needs into broad groups and arranged them in an

ascending sequence. When we fulfil our needs at the lowest level, the next level will dominate our attention.

The auctioneer can just as well cry out an

ascending sequence of prices to the gathered bidders, stopping the auction when enough are willing to take down the total issue.

A serial master file contains customer name and address records, arranged in

ascending sequence by customer number.

each

ascending sequence has an upper bound (modulo ([less than or equal to]))

Therefore, in case of this sport disciplines, the raport between alternating

ascending sequences with flat and descending, the diversity of running surfaces consistencies, as well as the variety of climatic conditions, determine the profile on which the athlet should gain progresively, through a guided action, specialised.