Adding name to registration form

Hi

I followed http://docs.thememylogin.com/adding-extra-registration-fields/ to add the first and last name to the registration form. I added the register-form.php pag with the correct code and the fields show – great.

But, they aren’t saving. I created the theme-my-login-custom.php file and added

<?php

function tml_registration_errors( $errors ) {
	if ( empty( $_POST['first_name'] ) )
		$errors->add( 'empty_first_name', '<strong>ERROR</strong>: Please enter your first name.' );
	if ( empty( $_POST['last_name'] ) )
		$errors->add( 'empty_last_name', '<strong>ERROR</strong>: Please enter your last name.' );
	return $errors;
}
add_filter( 'registration_errors', 'tml_registration_errors' );

function tml_user_register( $user_id ) {
	if ( !empty( $_POST['first_name'] ) )
		update_user_meta( $user_id, 'first_name', $_POST['first_name'] );
	if ( !empty( $_POST['last_name'] ) )
		update_user_meta( $user_id, 'last_name', $_POST['last_name'] );
}
add_action( 'user_register', 'tml_user_register' );

to it and saved in the plugins folder.

But new registrations aren’t showing a name.

Thanks for any help