end up this section by vs end up this section with

Both phrases are not commonly used in English. The correct way to express this idea is to use 'end this section with' or 'end this section by.' The phrases 'end up this section by' and 'end up this section with' are not grammatically correct and should be avoided.

Last updated: March 19, 2024

end up this section by

This phrase is not correct in English.

This phrase is not commonly used in English. To express the idea of concluding a section, it is better to use 'end this section by.'

Show examples from the web [+]

We end up this section by remarking that conversely, every Killing vector field on the round sphere (and all the more on its quotients) satisfies (14). This follows ...
We end up this section by deriving some useful algebraic relations satisfied by R+. Lemma 2.2. On any manifold we have. R+ ∧ I = I ∧ R+ + R, or, equivalently, ...
This is known as the propagation rules of quantum code constructions. We end up this section by presenting two examples to illustrate the previous construction.
We end up this section by the following proposition which will be used in the proof of Theorem. 4. It may be of independent interest since it sheds more light on ...

Alternatives:

end this section by
conclude this section by
finish this section by
wrap up this section by
complete this section by

end up this section with

This phrase is not correct in English.

This phrase is not commonly used in English. To convey the idea of finishing a section, it is more appropriate to use 'end this section with.'

Show examples from the web [+]

Let us end up this section with a direct application of Theorem 4.11. Corollary 4.12. Let E be a metric space together with its Borel σ-field. Bor(E). If P and P are ...
Lastly,)we)end)up)this)section)with)the)following)useful)theorem)without) rigorous)proof.) (14))Theorem.))Let' X 'be'irreducible'with'a'standard'semigroup'{ Pt } ...
Nov 12, 2015 ... We end up this section with a definition that includes the standing assumptions on the Schottky groups considered in mostly all of our ...
Let us end up this section with: Theorem 2 If ∆(D) > 0 one can put D(v) = -(v-v0)( v-v1)(v-v2) with v0 < v1 < v2; the superintegrable systems 그1 and 그2 given by ...

Alternatives:

end this section with
conclude this section with
finish this section with
wrap up this section with
complete this section with

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end up this section by vs end up this section with