teorema de zero de funció continua

teorema:
Si ( f(x) és continua & f(a) < 0 < f(b) ) ==> [∃c][ f(c) = 0 ]


Demostració:
[∀s][ s > 0 ==> lim |f(a_{n}) + (-1)f(c)| < s ]
[∀s][ s > 0 ==> lim |f(b_{n}) + (-1)f(c)| < s ]


lim f(a_{n}) = f(c)
lim f(b_{n}) = f(c)


f(a_{n}) [< 0 [< ( (f(a_{n})+f(b_{n}))/2 ) or ( (f(a_{n})+f(b_{n}))/2 ) [< 0 [< f(b_{n})


f(c) [< 0 [< ( (f(c)+f(c))/2 ) or ( (f(c)+f(c))/2 ) [< 0 [< f(c)