Exercise 1.13 in Section 1.2.2 asks for a proof that is the closest integer to , where .
Proof Idea: To show that is the closest integer to we simply need to show that for all . We will establish the result by proving two lemmas.
Lemma 1: for all .
Lemma 2: for all .
Notice that if Lemma 1 and Lemma 2 are true it will follow that for all .
It remains to be shown that Lemma 1 and Lemma 2 are indeed true.
Proof of Lemma 1: We will prove the result by Induction on . When n = 0 and n = 1, the result follows trivially. Let and suppose the result holds for and . Consider, . By definition, . Then by using the Induction Hypothesis, it follows that
which can be further reduced to
Since and , the result holds for . Therefore, by Induction, for all .
Proof of Lemma 2: implies that , i.e. . Also, implies that which implies that . So, for any . But, . Hence, for all and it trivially follows that for all . Since and decreases as increases, we can conclude that for all .
As planned, this completes the proof that is the closest integer to .