Plugin settings not saving

I’m having a devil of a time getting my plugin settings to save. I’ve gone through the code so many times I have to think I’ve simply missed something. I’m following the Setting API, but any changes I make on the plugin settings page don’t save. What did I miss?

function __construct() {
self::$instance = $this;

//  Add "Settings" link to plugin page
add_filter( 'plugin_action_links_' . plugin_basename( __FILE__ ) , array( $this, 'settings_link' ) );

// Creates the CEMB Seminar submenu
add_action( 'admin_menu', array( $this, 'add_menu' ) );

// Register and define the settings
add_action( 'admin_init', array( $this, 'settings_init' ) );

} // End of __construct()


/***** Menus *****/

//  Add "Settings" link to plugin page
function settings_link( $links ) {
    $settings_link = sprintf( '<a href="%s">%s</a>', admin_url( 'options-general.php?page=cemb-seminar' ), __( 'Settings' ) );
    array_unshift( $links, $settings_link );
    return $links;
}

// Creates the CEMB Seminar submenu
function add_menu() {
    add_submenu_page( 
        'edit.php?post_type=cemb_seminar', 
        __( 'Course Requirements' ), 
        __( 'Course Requirements' ), 
        'manage_options', 
        'cemb-seminar', 
        array( $this, 'settings_page' )
    );
}

/***** Course Requirement Settings *****/

/* Creates new database fields */
function settings_init() {
    $options = array(
        'cemb_seminar_seminars_college' => '0',
        'cemb_seminar_showcases_college' => '0',
        'cemb_seminar_writers_college' > '0',
        'cemb_seminar_aet_college' => '0',
        'cemb_seminar_mbu_college' => '0',
        'cemb_seminar_eis_college' => '0',
        'cemb_seminar_sng_college' => '0',
    );
    update_option( 'cemb_seminar_options', $options );


    register_setting( 
        'cemb_seminar_options', 
        'cemb_seminar_options' 
    );
    add_settings_section( 
        'cemb_seminar_reqs_college', 
        'Course Requirements', 
        array( $this, 'reqs_college_fn' ), 
        'cemb-seminar'
    );
    add_settings_field( 
        'cemb_seminar_seminars_college_field', 
        'Seminars', 
        array( $this, 'seminars_college_fn' ), 
        'cemb-seminar', 
        'cemb_seminar_reqs_college' 
    );
    add_settings_field( 
        'cemb_seminar_showcases_college_field', 
        'Showcases', 
        array( $this, 'showcases_college_fn' ), 
        'cemb-seminar', 
        'cemb_seminar_reqs_college' 
    );
    add_settings_field( 
        'cemb_seminar_writers_college_field', 
        'Writer's Nights', 
        array( $this, 'writers_college_fn' ), 
        'cemb-seminar', 
        'cemb_seminar_reqs_college' 
    );
    add_settings_field( 
        'cemb_seminar_aet_college_field', 
        'AET Seminars', 
        array( $this, 'aet_college_fn' ), 
        'cemb-seminar', 
        'cemb_seminar_reqs_college' 
    );
    add_settings_field( 
        'cemb_seminar_mbu_college_field', 
        'MBU Seminars', 
        array( $this, 'mbu_college_fn' ), 
        'cemb-seminar', 
        'cemb_seminar_reqs_college' 
    );
    add_settings_field( 
        'cemb_seminar_eis_college_field', 
        'EIS Seminars', 
        array( $this, 'eis_college_fn' ), 
        'cemb-seminar', 
        'cemb_seminar_reqs_college' 
    );
    add_settings_field( 
        'cemb_seminar_sng_college_field', 
        'SNG Seminars', 
        array( $this, 'sng_college_fn' ), 
        'cemb-seminar', 
        'cemb_seminar_reqs_college' 
    );

} // End of settings_init()

// Add note about the course requirements settings
function reqs_college_fn() {
    echo '<p>The course requirement options below will apply to all students for all majors.</p>';
}

// Checks for the selected value in the drop menus
function course_req_selected( $selected, $option ) {

    // Get options first
    $options = get_option( 'cemb_seminar_options' );

    // Check if the option matches the input
    if ( $options['$option'] == $selected ) {
        echo ' selected="selected"';
    }   
} // End of course_req_selected()

// Add college-wide Seminars requirements Field
function seminars_college_fn() {

    // Get options from database
    $options = get_option( 'cemb_seminar_options' );
    $option = $options['cemb_seminar_seminars_college'];

    // Build the select form ?>
    <select id="cemb_seminar_seminars_college" class="cemb_seminar_settings_field" name="cemb_seminar_options[cemb_seminar_seminars_college]">
        <option value="0" <?php if ( $option == '0' ) { echo 'selected="selected"'; } ?>>0</option>
        <option value="1" <?php if ( $option == '1' ) { echo 'selected="selected"'; } ?>>1</option>
        <option value="2" <?php if ( $option == '2' ) { echo 'selected="selected"'; } ?>>2</option>
        <option value="3" <?php if ( $option == '3' ) { echo 'selected="selected"'; } ?>>3</option>
        <option value="4" <?php if ( $option == '4' ) { echo 'selected="selected"'; } ?>>4</option>
        <option value="5" <?php if ( $option == '5' ) { echo 'selected="selected"'; } ?>>5</option>
    </select> <?php

} // End of seminars_college_fn()

// Add college-wide Showcases requirements Field
function showcases_college_fn() {

    // Get options from database
    $options = get_option( 'cemb_seminar_options' );
    $option = $options['cemb_seminar_showcases_college'];

    // Build the select form ?>
    <select id="cemb_seminar_showcases_college" class="cemb_seminar_settings_field" name="cemb_seminar_options[cemb_seminar_showcases_college]">
        <option value="0" <?php if ( $option == '0' ) { echo 'selected="selected"'; } ?>>0</option>
        <option value="1" <?php if ( $option == '1' ) { echo 'selected="selected"'; } ?>>1</option>
        <option value="2" <?php if ( $option == '2' ) { echo 'selected="selected"'; } ?>>2</option>
        <option value="3" <?php if ( $option == '3' ) { echo 'selected="selected"'; } ?>>3</option>
        <option value="4" <?php if ( $option == '4' ) { echo 'selected="selected"'; } ?>>4</option>
        <option value="5" <?php if ( $option == '5' ) { echo 'selected="selected"'; } ?>>5</option>
    </select> <?php

} // End of showcases_college_fn()

// Add college-wide Writer's Night requirements field
function writers_college_fn() {

    // Get options from database
    $options = get_option( 'cemb_seminar_options' );
    $option = $options['cemb_seminar_writers_college'];

    // Build the select form */?>
    <select id="cemb_seminar_writers_college" class="cemb_seminar_settings_field" name="cemb_seminar_options[cemb_seminar_writers_college]">
        <option value="0" <?php if ( $option == '0' ) { echo 'selected="selected"'; } ?>>0</option>
        <option value="1" <?php if ( $option == '1' ) { echo 'selected="selected"'; } ?>>1</option>
        <option value="2" <?php if ( $option == '2' ) { echo 'selected="selected"'; } ?>>2</option>
        <option value="3" <?php if ( $option == '3' ) { echo 'selected="selected"'; } ?>>3</option>
        <option value="4" <?php if ( $option == '4' ) { echo 'selected="selected"'; } ?>>4</option>
        <option value="5" <?php if ( $option == '5' ) { echo 'selected="selected"'; } ?>>5</option>
    </select> <?php

} // End of writers_college_fn()

// Add college-wide AET Seminars requirements field
function aet_college_fn() {

    // Get options from database
    $options = get_option( 'cemb_seminar_options' );
    $option = $options['cemb_seminar_aet_college'];

    // Build the select form ?>
    <select id="cemb_seminar_aet_college" class="cemb_seminar_settings_field" name="cemb_seminar_options[cemb_seminar_aet_college]">
        <option value="0" <?php if ( $option == '0' ) { echo 'selected="selected"'; } ?>>0</option>
        <option value="1" <?php if ( $option == '1' ) { echo 'selected="selected"'; } ?>>1</option>
        <option value="2" <?php if ( $option == '2' ) { echo 'selected="selected"'; } ?>>2</option>
        <option value="3" <?php if ( $option == '3' ) { echo 'selected="selected"'; } ?>>3</option>
        <option value="4" <?php if ( $option == '4' ) { echo 'selected="selected"'; } ?>>4</option>
        <option value="5" <?php if ( $option == '5' ) { echo 'selected="selected"'; } ?>>5</option>
    </select> <?php

} // End of aet_college_fn()

// Add college-wide MBU Seminars requirements field
function mbu_college_fn() {

    // Get options from database
    $options = get_option( 'cemb_seminar_options' );
    $option = $options['cemb_seminar_mbu_college'];

    // Build the select form ?>
    <select id="cemb_seminar_mbu_college" class="cemb_seminar_settings_field" name="cemb_seminar_options[cemb_seminar_mbu_college]">
        <option value="0" <?php if ( $option == '0' ) { echo 'selected="selected"'; } ?>>0</option>
        <option value="1" <?php if ( $option == '1' ) { echo 'selected="selected"'; } ?>>1</option>
        <option value="2" <?php if ( $option == '2' ) { echo 'selected="selected"'; } ?>>2</option>
        <option value="3" <?php if ( $option == '3' ) { echo 'selected="selected"'; } ?>>3</option>
        <option value="4" <?php if ( $option == '4' ) { echo 'selected="selected"'; } ?>>4</option>
        <option value="5" <?php if ( $option == '5' ) { echo 'selected="selected"'; } ?>>5</option>
    </select> <?php

} // End of mbu_college_fn()

// Add college-wide EIS Seminars requirements field
function eis_college_fn() {

    // Get options from database
    $options = get_option( 'cemb_seminar_options' );
    $option = $options['cemb_seminar_eis_college'];

    // Build the select form ?>
    <select id="cemb_seminar_eis_college" class="cemb_seminar_settings_field" name="cemb_seminar_options[cemb_seminar_eis_college]">
        <option value="0" <?php if ( $option == '0' ) { echo 'selected="selected"'; } ?>>0</option>
        <option value="1" <?php if ( $option == '1' ) { echo 'selected="selected"'; } ?>>1</option>
        <option value="2" <?php if ( $option == '2' ) { echo 'selected="selected"'; } ?>>2</option>
        <option value="3" <?php if ( $option == '3' ) { echo 'selected="selected"'; } ?>>3</option>
        <option value="4" <?php if ( $option == '4' ) { echo 'selected="selected"'; } ?>>4</option>
        <option value="5" <?php if ( $option == '5' ) { echo 'selected="selected"'; } ?>>5</option>
    </select> <?php

} // End of eis_college_fn()

// Add college-wide SNG Seminars requirements field
function sng_college_fn() {

    // Get options from database
    $options = get_option( 'cemb_seminar_options' );
    $option = $options['cemb_seminar_sng_college'];

    // Build the select form ?>
    <select id="cemb_seminar_sng_college" class="cemb_seminar_settings_field" name="cemb_seminar_options[cemb_seminar_sng_college]">
        <option value="0" <?php if ( $option == '0' ) { echo 'selected="selected"'; } ?>>0</option>
        <option value="1" <?php if ( $option == '1' ) { echo 'selected="selected"'; } ?>>1</option>
        <option value="2" <?php if ( $option == '2' ) { echo 'selected="selected"'; } ?>>2</option>
        <option value="3" <?php if ( $option == '3' ) { echo 'selected="selected"'; } ?>>3</option>
        <option value="4" <?php if ( $option == '4' ) { echo 'selected="selected"'; } ?>>4</option>
        <option value="5" <?php if ( $option == '5' ) { echo 'selected="selected"'; } ?>>5</option>
    </select> <?php

} // End of sng_college_fn()

// Creates the course requirements page
function settings_page() {
    ?>
    <div class="wrap">
    <div class="icon32" style="background-image:url(<?php echo WP_PLUGIN_URL . '/cemb-seminar/images/cemb-logo32x32.png'; ?>); background-repeat:no-repeat;"><br /></div>
    <h2>Course Requirements</h2>
    <form method="post" action="options.php">
    <?php settings_fields( 'cemb_seminar_options' ); ?>
    <?php do_settings_sections( 'cemb-seminar' ); ?>
    <br />
    <input class="button-primary" type="submit" name="Submit" value=" <?php _e( 'Save Settings' ); ?> " />
    </form>
    </div> <?php

} // End of settings_page()