All we have to do is set an AI towards self improvement/production of better AIs.
All we have to do!!! - We have absolutely no idea how to do this, so how could a AI do this if we don't know how to do this?
Chicken and egg without a chicken or egg or anything remotely like either.
'set an AI towards self improvement' That's like asking a rabbit to design and manufacture a better rabbit - the rabbit would not comprehend what you are asking of it.