How do I keep the variable value in a MySQL stored procedure from changing when records are updated?

SELECT @myName := username FROM User WHERE ID=1;

UPDATE User SET username = 'Fred' WHERE ID=1;

SELECT username INTO myName FROM User WHERE ID=1;

drop table if exists user;
create table user (
  id serial primary key,
  username varchar(10)
);
insert into user (username) values ('Paul');

drop procedure if exists doit;
delimiter !!
create procedure doit()
begin
  declare name varchar(10);
  select @name:=username from user where id=1;
  select @name; -- shows 'Paul' as expected
  update user set username = 'Fred' where id=1;
  select @name; -- still shows 'Paul'
end!!
delimiter ;

call doit();