Translation of the written part:

Exercise

(a) Induction from any natural Try the following result which is very useful when we want to try that some property is valid from some natural not necessarily 0 or 1. Here N contains the zero. Be (1) such that (i) (2) (ii) (3) Show that (4) Tip: Induction in m.

(b) Be (5) a limited set upwards. Show that X has maximum. Tip: Use the principle of good order in Z.

(c) Be A, B, C and D real numbers not negative such that: (i) (6) (ii) (7) (iii) (8) Show that (9).

Note: I kindly ask you to use tools for the rigorous demonstration that is convenient with the beginning of the programmatic content of the Real Analysis course. Please specify the steps.

I would appreciate a detailed resolution.

Exerccio (a) Induo a partir de um natural qualquer Prove o seguinte resultado que muito til para quando queremos provar que alguma propriedade vlida a partir de algum natural no necessariamente 0 ou 1. Aqui N contm o zero. Seja X CN tal que (i) a E X (2) (4) (ii) ne X+n+1 X. (3) Mostre que {a, a +1, a + 2, ...} = {a + m;m e N} CX. Dica: Induo em m. (b) Seja X Z um conjunto limitado superiormente. Mostre que X tem mximo. Dica: Use o princpio da boa ordenao em Z. (c) Sejam A, B, C e D nmeros reais no negativos tais que: i)A