If I have a textarea block like so...
Nope! If you validated that in an HTML document, you'd get an error along the lines of
document type does not allow element "b" here
What you might be looking for is a What-You-See-Is-What-You-Get editor, which is actually an which is making use of a JavaScript feature designMode. It's possible to find tutorials on how to create these (archive), but it's a much, much better idea to use an existing one, as for it to be really useful and usable takes a lot of work.
Take note that - if you were still interested in validation, which you should be if you're working with HTML - is that you won't be able to use a strict DOCTYPE declaration, as the WYSIWYG editor will be using an iframe.