Using the range operator (1..$end) is the way to go, but another (less desireable) variation uses a C-style loop:

#!/usr/bin/perl
use strict;
use warnings;

my $output = "1 ";

print "Enter a positive integer: ";

my $lastnum = <STDIN>;
# chomp $lastnum;                      # not needed here
if ( $lastnum =~ /^[0-9]+$/ ) {        # TEST, don't simply trust, use
+r input!
   my $n = 2;
   while ( $n < ($lastnum+1) ) {
       $output .= $n;
       $output .= " ";
       $n++;
    }
    print "$output \n";
} else {
   print  "Bad input\n";
   exit;
}
[download]

Runs thus:

ww@GIG:~/pl_test$ perl 675693.pl
Enter a positive integer: -3
Bad input
ww@GIG:~/pl_test$ perl 675693.pl
Enter a positive integer: 27
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 2
+7
ww@GIG:~/pl_test$ perl 675693.pl
Enter a positive integer: 1.35
Bad input
ww@GIG:~/pl_test$
[download]