## Constructive real numbers and constructive function spaces by N. A. Sanin

By N. A. Sanin

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Then We deduce that Proof (i)It is sufficient to note that x=[x]+{x} with 0⩽{x}<1. (ii)Since [x+n]=x+n+θ 1 and [x]=x+θ 2 with −1<θ i ⩽0, we have so that |[x+n]−([x]+n)|<1 and we conclude by noting that [x+n]−([x]+n)∈ℤ. For the second equality, we have (iii)Using (ii) we have on the one hand On the other hand, if x=[x]+θ 1 and y=[y]+θ 2 with 0⩽θ i <1, then we have since 0⩽θ 1+θ 2<2 implies [θ 1+θ 2]=0 or 1. (iv)∑ n⩽x 1=[x] if x⩾1 by convention. If 0⩽x<1, then ∑ n⩽x 1=0. (v)If 0⩽x<1, then there is no multiple of d which is ⩽x and [x/d]=0 in this case.

Suppose the result is true with k replaced by k−1. We use (i) applied to each interval [x i ,x i+1] which implies that F′ possesses k zeros y i such that x i

Setting and using the first identity above, we therefore get and we conclude the proof with . 8 Exercises 1 Let a, b be positive integers. In the Euclidean division of a by b, the quotient q and the remainder r satisfy r⩾q. Show that, in the Euclidean division of a by b+1, we get the same quotient. 2 Let a, q be positive integers. We denote by the set of positive integers b such that q is the quotient of the Euclidean division of a by b. Show that 3 Let m,n∈ℤ∖{0}. Show that (i) . (ii)If m∤n, then .