wordpress ajax pagination

I’m working on a WordPress Theme (create a custom theme) and using AJAX as pagination for my Custom Posts.

So I’m going to load pages with AJAX but the problem is when I click for next page, instead of next page it’s returning the home page.

Now I’m going to make it easier with a simple sample, if you can help me to call HELLO query in sample below, I will fix the whole codes with your path.

Here is my HELLO target for example (we should call it):

This is my code in functions.php

add_action("wp_ajax_check_user", "check_user");
add_action("wp_ajax_nopriv_check_user", "check_user");

function check_user() {

    echo 'hello';
    die();
}

And this is my code to call “HELLO” in some page:

$.ajax({
        type: 'POST',
        url: "<?php echo admin_url('admin-ajax.php') ?>",
        data: { 
            'action': 'check_user'
            },
        success : function(data) {
            alert(data); //FOR DEBUG
        }

    });

So I can’t call hello and instead I see my home page.

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UPDATE:
@face

this is my sample page and you can see the results after click on next page or page 2:

here is my sample page

at the bottom of the page please try to go next page or page 2. i made the results as a message box.