[see also: just
If we simply mimic the standard proof of...... we are led to......
In case M is empty we write simply f≈ g.
The rank of appearance (or simply the rank) of an integer in the sequence U(P) is the least positive integer n such that......
The lower limit is defined analogously: simply interchange sup and inf in (1).
Then G is simply g with its periodic string read backwards.
The reason for preferring (1) to (2) is simply that (1) is manifestly invariant. [Note the double r in preferring.]