do not dup in category foreach loop

<?php
//for each category, show 3 posts
$cat_args=array(
  'orderby' => 'name',
  'order' => 'ASC'
   );

$do_not_duplicate = array();

$categories=get_categories($cat_args);
  foreach($categories as $category) {
    $args=array(
      'showposts' => 3,
     'post__not_in' => $do_not_duplicate,
      'category__in' => array($category->term_id),
      'caller_get_posts'=>1
    );
    $posts=get_posts($args);
      if ($posts) {
        echo '<p>Category: <a href="' . get_category_link( $category->term_id ) . '" title="' . sprintf( __( "View all posts in %s" ), $category->name ) . '" ' . '>' . $category->name.'</a> </p> ';
        foreach($posts as $post) {
          setup_postdata($post);
    $do_not_duplicate[] = $post->ID; ?>
          <p><a href="<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title_attribute(); ?>"><?php the_title(); ?></a></p>
          <?php
        } // foreach($posts
      } // if ($posts
    } // foreach($categories
?>