Too many sessions open = 'max_user_connections' active connections

Hi,
On my site I have told by my host that too many session are remaining opne and I get the below error in the error log file:

mysql_pconnect() [<a href='function.mysql-pconnect'>function.mysql-pconnect</a>]: User a_db has already more than 'max_user_connections' active connections in /home/dir/public_html/l/adodb/adodb-mysql.inc.php on line 85

I have checked that file and it has the following code is the below okay?:

// returns true or false
function _connect($argHostname, $argUsername, $argPassword, $argDatabasename)
{
$this->_connectionID = mysql_connect($argHostname,$argUsername,$argPassword);
if ($this->_connectionID === false) return false;
if ($argDatabasename) return $this->SelectDB($argDatabasename);
return true; 
}

// returns true or false
function _pconnect($argHostname, $argUsername, $argPassword, $argDatabasename)
{
$this->_connectionID = mysql_pconnect($argHostname,$argUsername,$argPassword);
if ($this->_connectionID === false) return false;
if ($argDatabasename) return $this->SelectDB($argDatabasename);
return true; 
}

PHP:

Also I have checked the normal php files and they have :
session_start();
in the beginning but no "session_write_close();"

Do I need to add this at the end of each file?