end

1

[see also: aim, purpose]

At the end of Section 2, we prove ...... [Not: “In the end of”]

Now $F$ is defined to make $G$ and $H$ match up at the left end of $I$.

We shall find it convenient not to distinguish between two such sequences which differ only by a string of zeros at the end.

To this end, we first consider ...... [= For this purpose; not: “To this aim”]

Thus in the end $F$ will be homogeneous. [= finally]

2

[see also: conclude, finish, terminate]

The path ends at $x$.

The word ends in $a$.

a word starting with $a$ and ending with $b$

The exact sequence ends on the right with $H(X)$.

We end this section by stating without proof an analogue of ......

Arguing as before, we shall end up with a simple tree all of whose facets contain $V$.



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