Display 5 latest posts in each category in WordPress

<div id="page-not-found" class="post-page">

<?php
$cat_args = array(
  'orderby' => 'name',
  'order' => 'ASC',
  'child_of' => 0
);

$categories =   get_categories($cat_args); 

foreach($categories as $category) {
    echo '<dl>';
    echo '<dt> <a href="' . get_category_link( $category->term_id ) . '" title="' . sprintf( __( "View all posts in %s" ), $category->name ) . '" ' . '>' . $category->name.'</a></dt>';

     $post_args = array(
      'numberposts' => 5,
      'category' => $category->term_id
    );

    $posts = get_posts($post_args);

 foreach($posts as $post) {
 ?>
  <dd><a href="<?php the_permalink(); ?>"><?php the_title(); ?></a></dd>
  }
 echo '<dd class="view-all"> <a href="' . get_category_link( $category->term_id ) . '" title="' . sprintf( __( "View all posts in %s" ), $category->name ) . '" ' . '>View all posts in ' . $category->name.'</a></dd>';
 echo '</dl>';
}
?>
<div class="page-end"></div>

</div>

CSS

#page-not-found dl {
width: 200px;
float: left;
padding: 0 18px 0 0;
height: 250px;
}

#page-not-found dt {
font-weight: bold;
font-size: 1.1em;
padding: 10px 0;
}

#page-not-found dt a {
color: #b0bf32;
}

#page-not-found dd.view-all {
border-top: 1px solid #c9c9c9;
font-size: 0.9em;
margin: 5px 0;
padding: 2px 0 0;
text-align: right;
}

#page-not-found dd.view-all a {
color: #999;
}

.page-end {
clear: both;
}