Questão de Raciocínio Lógico

ŝŶƚĞŐƌĂĕĆŽĚĞĨƵŶĕƁĞƐĠƵŵĚŽƐƉƌŝŶĐŝƉĂŝƐƚſƉŝĐŽƐĚĂŶĄůŝƐĞĞƐƐĞŶĐŝĂůŵĞŶƚĞƚĞſƌŝĐĂ͕ĞŶƋƵĂŶƚŽĂƐĞƋƵĂĕƁĞƐ ĚŝĨĞƌĞŶĐŝĂŝƐƉŽƐƐƵĞŵĚŝǀĞƌƐĂƐĂƉůŝĐĂĕƁĞƐĚĞŶƚƌŽĞĨŽƌĂĚĂDĂƚĞŵĄƟĐĂ͘ŵĂŵďŽƐŽƐĐĂƐŽƐ͕ŽĞƐƚƵĚŽĚĂƐ ƐĞƋƵġŶĐŝĂƐĞƐĠƌŝĞƐĚĞĨƵŶĕƁĞƐĐŽŵƌĞůĂĕĆŽĂĐŽŶǀĞƌŐġŶĐŝĂ͕ĚĞƌŝ ǀĂĕĆŽĞŝŶƚĞŐƌĂĕĆŽƚĞƌŵŽĂƚĞƌŵŽ͕ŽĐƵƉĂƵŵ ƉĂƉĞůĞƐƐĞŶĐŝĂů͘ŝĂŶƚĞĚŝƐƐŽ͕ĂŶĂůŝƐĞĂƐĂĮƌŵĂĕƁĞƐĂƐĞŐƵŝƌ͘ xn /͘ ŽŶƐŝĚĞƌĞĂƐĞƋƵġŶĐŝĂĚĞĨƵŶĕƁĞƐ fn : [0, 1] → R͕ĚĂĚĂƉŽƌ f (x) = ƉĂƌĂƚŽĚŽn = 1, 2, . . . ∑∞ n n! ^Ğf(x) = fn(x)͕ĞŶƚĆŽ f(x) = 0 ƉĂƌĂƚŽĚŽx ∈ [0, 1]. n=1 //͘ ŽŶƐŝĚĞƌĞ J = [0, 1] e g : J → R ĚĂĚĂ ƉŽƌ { 1 se x ∈ Q ∩ J g(x) = 0 se x ∈ Q ∩ J KďƐĞƌǀĞ ƋƵĞ͘ Q ∩ J ∫ 1 ƚĞŵŵĞĚŝĚĂŶƵůĂ͕ŽƵƐĞũĂ͕ g(x) = 0 ĞŵƋƵĂƐĞƚŽĚŽƉŽŶƚŽĚĞJ͘>ŽŐŽ͕ g(x)dx = 0 . 0 xn ///͘ ƐĞƋƵġŶĐŝĂ ĚĞ ĨƵŶĕƁĞƐ fn : [0, 1] → R͕ ĚĂĚĂ ƉŽƌ f (x) = ƉĂƌĂ ƚŽĚŽ n = 1, 2, . . . ƐĂƟƐĨĂnj ∑∞ ∫ 1 ∫ 1 ∑∞ fn(x)dx = fn(x)dx . n=1 0 0 n=1 ĐŽƌƌĞƚŽŽƋƵĞƐĞĂĮƌŵĂĞŵ