[see also: condition, need]
We can assume, by decreasing $n$ if necessary, that ......
Passing to a subsequence if necessary, we can assume that ......
We can assume that $p$ is as close to $q$ as is necessary for the following proof to work.
This is necessary for determining the constants in Theorem 2.
Although these proofs run along similar lines, there are subtle adjustments necessary to fit the argument to each new situation.
It is therefore unnecessary to specify $G$ on $ M$.
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