Return False not working inside jQuery.ajax

Question

P.S.: Read \"EDITED on 2019-06-29\":

I have a webform for updating user information, and when he updates his email a verification is performed via

Answer 1

$('form').submit(function() {
// we send an AJAX request to validate the unicity of the email
$.ajax({
    url: '/ajax/verify-email.php',
    type: 'POST',
    data: { email: $('#email').val() },
    // we set the context to the form so that inside
    // the success callback 'this' points to the form
    context: this,
    success: function(result) {
        if (result != '1') {
            // If the server send something different than 1
            // we know that the email is unique and trigger
            // the submission of the form using the underlying
            // DOM element to avoid calling the .submit handler
            // recusrively
            this.submit();
        } else {
            // The email is not unique => we are informing
            // the user that the email is already in use
            alert('Another user is using this email');
            $('#email').focus();
        } 
    }
});

// we cancel the normal submission of the form    
return false;

});