Sentence examples for conclude the sequence from inspiring English sources

To conclude the sequence, the reader may refer to the article on human emergence that I published in this journal last year (Tattersall 2009; doi 10.1007/s12052-009-0614x); butheor the sake of both completeness and of currency in this issue, the principal points of that article are summarized below and its argument is updated.

We conclude that the sequence ({x_{2}(n)}) is nondecreasing.

We conclude that the sequence ({Tx_{n}}) is Cauchy.

We conclude that the sequence ({x_{n}}) is Cauchy.

Hence, we conclude that the sequence { x n } is Cauchy.

So, from the above discussion, we can conclude that the sequence {x n −q}n≥1 is bounded.

Thus, we conclude that the sequence { d ( x n, x n + 1 ) } is nonnegative and nonincreasing.

Hence, by Lemma 2.3, we conclude that the sequence converges strongly to.

Then from Theorem 3.3 we conclude that the sequence ({x_{n}}) converges weakly to 1.

end{aligned} Thus, we conclude that the sequence ((x_{n})) is bounded.

Thus, we conclude that the sequence of nonnegative reals is bounded.