Use a hash. Something like the following

# opening of files etc.

my %users = ();


while ($line = <errlog>) {
    chomp $line;
    next unless ($line =~ /Login succeeded/);
    $users{(split / /, $line)[6]} ||= 1;
}

foreach (sort keys %users) {
   print LOGINF $_, "\n";
}

# closing fhs etc
[download]

NB: consider using lexical filehandles.

hth,