An infinite while loop will work:

use warnings;
use strict;

print "Please give me a number:\n";
my $initial_number = <STDIN>;
chomp $initial_number;

while (1) {
    print "Please continue with a new number:";
    my $new_number = <STDIN>;
    chomp $new_number;
    if ($new_number < $initial_number) {
        print "You specified a number which is smaller than the previo
+us. The program will now exit\n";
        exit;
    }
    $initial_number = $new_number;
}
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