i have in my lecture notes that complete is not equal to close,,, what does it mean ?

and also i have this

"Because with completeness, convergent infinite sums are defined it is possible to introduce smaller basis than the Hamel basis "

as i know , when a general sentence of sequence is convergent to zero then it is possible that the series composed of that be convergent but according the above sentences completeness make a series convergent ,, why?

and the other question why when the series converges it can introduce for us smaller basis than Hamel basis ?