Let's say you want from >lmo0025 to >lmo0026 inclusive. If you know that the next record will be >lmo0027 you can do something like this:

use strict;
while (<>) {
    if (/^>lmo0025/ .. /^>lmo0027/) {
        last if /^>lmo0027/; 
        print;
    }
}
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If you don't know that the next record is >lmo0027 you can do something like this:

use strict;
my ($firstfoundflag, $lastfoundflag);
while (<>) {
    if ($firstfoundflag || /^>lmo0025/) {
        $firstfoundflag++;
           if ($lastfoundflag || /^>lmo0026/ ) {
            $lastfoundflag++;
            if (/^>lmo(?!0026)/) { 
                last; 
            }
        }
        print;
    }
}
[download]