不确定sql或php是否有错误

I am currently following a tutorial on Youtube called Register & Login/PHP tutorials by Alex from Phpacademy.. am in part 5 and here is login.php

<?php
include 'core/init.php';

if (empty($_POST) === false) { 

    $username = $_POST['username']; 
    $password = $_POST['password']; 

    if (empty($username) === true || empty($password) ===   true) {

        $errors[]  = 'You need to enter a username and password ';

    } else if (user_exists($username) === false) {
        $errors[] = 'We couldn\'t find that username. Have you registered?';
    } 
    else if (user_active($username) === false){
        $errors[] = 'You havn\'t activated your account!';  
    } 
    else  { 
        $login = login($username, $password);
        if ($login === false) {
            $error[] = 'That username/password combination is incorrect';
        } else {
        $_SESSION['user_id'] = $login;
        header('Location: index.php');
        exit();
        }
    } 

}

print_r($errors);

?>

Here is users.php

<?php

function user_exists($username) { 
$username = sanitize($username); 
return (mysql_result(mysql_query("SELECT COUNT(`user_id`) FROM `users` WHERE `username` = '.$username'"), 0) == 1) ? true : false;
}

function user_active($username) { 
$username = sanitize($username); 
return (mysql_result(mysql_query("SELECT COUNT(`user_id`) FROM `users` WHERE `username` = '.$username'  AND `active` = 1 ") , 0 ) == 1 ) ? true : false;

}

function user_id_from_username($username){

    $username = sanitize($username);
    return mysql_result (mysql_query("SELECT `user_id` FROM `users` WHERE `username` = '$username' "), 0, 'user_id');

}
function login($username, $password){

    $user_id = user_id_from_username($username);
    $username = sanitize($username);
    $password = md5($password);
    return (mysql_result(mysql_query("SELECT COUNT(`user_id`) FROM `users` WHERE `username` = '.$username' AND `password` = '.$password'"), 0) == 1) ? $user_id : false;
}
?>

and here is the output Array ( [0] => We couldn't find that username. Have you registered? )

Am new here, apologies in advance

WHERE `username` = '.$username' AND `password` = '.$password'"

Remove the dots

Your SQL queries are going to be returning bad results. Otherwise, you will be searching for .jond in your database if the username they entered is jond.

return (mysql_result(mysql_query("SELECT COUNT(`user_id`) FROM `users` WHERE `username` = '.$username'"), 0) == 1) ? true : false;

Remove the . before $username and $password in the query.

"SELECT COUNT(`user_id`) FROM `users` WHERE `username` = '$username'"

Your query needs a tad bit tweaking. Remove the period in front of the username since it's inside the double quotes

return (mysql_result(mysql_query("SELECT COUNT(`user_id`) FROM `users` WHERE `username` = '$username'"), 0) == 1) ? true : false;

This goes the same for the other queries in that file. As mentioned in the comments, you really ought to switch from the deprecated mysql_* functions to PDO/mysqli so that your code will still work in future versions of PHP, and you won't be open to injection hacks.

Your code is pretty hideous overall. You should NOT be nesting your mysql calls like that. Nesting like that implies that you think a DB operation will NEVER fail. This is a VERY BAD assumption.

That being said, here's at least one source of your problems:

return (...snip ... WHERE `username` = '.$username'"), 0) == 1) ? true : false;
                                        ^--- here

You've embedded a . in that query, making all your usernames look like .foo instead of just foo. The problem exists in both user_exists(), user_active() AND login().