. a (ε-) Uε (a) = (a −ε, a + ε).
. (a n) a, :
limn→∞ a n = a ⇔∀ε > 0 ∃N ∀n ≥ N | a n − a | < ε.
, a (a n) N(ε), N, , ∀n ≥ N | a n − a | < ε ε.
. , n- a n =n/(2n+1) N(ε) = .
3. ∃limn→∞ a n = a ⇒∃M1, M2∀nM1£ a n£M2.
.∀ε > 0 ∃N(ε)∀n ≥ N
− ε< a n a <ε a − ε< a n< a + ε.
M1 = min(min(a 1,..., a N−1), a ε); M2 = max(max(a 1,..., a N−1), a + ε). ∀nM1£ a n£M2.
4. a, limn→∞ a n= a, a .
. .
∀ε > 0 ∃N1∀n ≥ N1 | a n − a | < ε/2
∀ε > 0 ∃N2∀n ≥ N2 | a n − b | < ε/2.
M = max(N1, N2). | a n− a |+ | a n− b | <ε, | a n − a | + | a n − b | ≥ | a − b |, | a − b | <ε. , ε .
,
: ∃limn→∞ a n = a, ∃limn→∞ b n = b. :
∀ε> 0 ∃N1∀n ≥ N1 | a n − a | <ε
∀ε> 0 ∃N2∀n ≥ N2 | b n − b | <ε
a n = a + αn
b n = b + βn