24. .

23.10.85

I,J∈N, N∈Z, X,Y,Z∈R.

1. ,
A) ∀>0 ∀Y>0 ∃N: (N-l)*X≤Y<N*X
B) ∀>0 ∀Y>0 ∃N: XN-1≤X<XN
C) INF{Y/I|Y>0;I∈N}=0

. ) ∈R - , R - ( ). ( ( ), ).
B) [X,Y] ⇐def⇒ {Z| X≤Z≤Y}
C) (X,Y) ⇐def⇒ {Z| X<Z<Y} (Y-X) ().

. () [XI,YI] () ⇐def⇒ I∈N: [XI,YI]⊂[XI-1,YI-1]

2. ) .
) 0 .

3. ) ⇔ ∀>0 ∃I: YI-XI<E.

. ) F: J→R
B) F: N→R.
C) " BCE"⇐def⇒"BCE, "
D) " 8"⇐def⇒", ".

.

. 2=1+1; 3=2+1; 4=3+1; 5=4+1; 6=5+1; 7=6+1; 8=7+1; 9=8+1; 10=9+1.
T = {0,1,2,3,4,5,6,7,8,9} - . I: i → N: I((i,i-1,...,1)) = N*10i-1 + N-1*10i-2 + ... + 1
(i,i-1,...,1) ii-1i-2...1 .

4, ) - I 0 ⇒ - ...?
B) . , .
C) , I 0, - -.

5. .

. W: Z⊗Z⊗(N\{9,99,999,...}∪{0})→Q W(k,n,a) = 10k*(n + a/10L(a)) L(a) - 10, a (L(0)=0)

6. . (k,n,a)

6. A) W - .
) . , , W - ,
c) : (|n|≡a (mod 10)) ⇒ L(|n|)=k) W -

. V: Z⊗{ , 9} R:
V((±b,1, 2, 3,...,N,...)) = ±sup{b+1*10-1+2*10-2+...+N*10-n}, N . Z⊗{} .

7. a) V - .
b)** , V - .

. [0, 1] 0,12...N...

.

. 01 - 0 1.

8. ( -), X , X. X ()

9. a) (N)
) 0101
c) R
d) [0,1]
e)
f)
g) R, 7

10. : ) b) ) d) T

11. , . ( .)

(-): , .

.

R : R = R ∪ {-∞, +∞}
:
) ∀x∈R: x+∞ = +∞; x-∞ = -∞ x/+∞ = x/-∞ = 0
) ∀x>0: x*(+∞)=+∞; x*(-∞)=-∞
) ∀x<0: x*(+∞)=-∞; x*(-∞)=+∞
D) .
: +, -, *, /, A=±∞, B=±∞ .

12. R (- 12 - 15)
R * .

13. R.