teorema del suprem y el ínfim

teorema:
Si ( A = [a,c]_{K} & (c,b)_{K} = B ) ==> [∃inf(B)][ inf(B) = c ]


Demostració:
lim max(a_{n}) = c
lim inf(b_{n}) = c


max(a_{n}) [< inf(B) [< ( (max(a_{n})+inf(b_{n}))/2 ) or ...
... ( (max(a_{n})+inf(b_{n}))/2 ) [< inf(B) [< inf(b_{n})


c [< inf(B) [< ( (c+c)/2 ) or ( (c+c)/2 ) [< inf(B) [< c


teorema:
Si ( A = (a,c)_{K} & [c,b]_{K} = B ) ==> [∃sup(A)][ sup(A) = c ]


Demostració:
lim sup(a_{n}) = c
lim min(b_{n}) = c


sup(a_{n}) [< sup(A) [< ( (sup(a_{n})+min(b_{n}))/2 ) or ...
... ( (sup(a_{n})+min(b_{n}))/2 ) [< sup(A) [< min(b_{n})


c [< sup(A) [< ( (c+c)/2 ) or ( (c+c)/2 ) [< sup(A) [< c