teorema de successió parcial convergent

teorema:
Si a < c_{n} < b ==> [∃c_{n_{k}}][ a < c_{n_{k}} < b & lim c_{n_{k}} = c ]


Demostració:
lim a_{n} = c
lim b_{n} = c


a_{n} [< c_{n_{k}} [< ( (a_{n}+b_{n})/2 ) or ( (a_{n}+b_{n})/2 ) [< c_{n_{k}} [< b_{n}


c [< c_{n_{k}} [< ( (c+c)/2 ) or ( (c+c)/2 ) [< c_{n_{k}} [< c