11. Diziler

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-+0-5(- ,=4=+ an 42$-)(2$-+-+ {a} &=)25- -?-+02 '()*5% 72$2&-$! >. -52
n ≥ no ⇒ an ∈ {a} ⇒ an = a
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A′ = {an : n ∈ N} #2,3&240&0- 40- <)*+ $&5'='%52%1 4' <)*+%0% ()*&+,+%0
2F3< 40- <)*+ $&*2&353-G ,2%0 A ∈ T 50-6 a ∈
/ A′ $&5'='%52%1 a ∈ A ⇒ A ∈
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v 1 ⊃ V2 ⊃ · · · ⊃ Vn ⊃ · · ·
40F0*0%5+ 0FB0F+ 40- <$D'&'<&2- (242%3 P,+-+& (242%Q >2-53-6 A = {an : n ∈
N} 50,+&0*6
x ∈ Ā ⇔ [(∀n)(Vn ∩ A 6= ∅)]
⇔ [(∀n)(∃kn )(ank ∈ Vn )]
⇔ [ank → x]
"6 E+-F+& #2,3&2-52% $&'D2% 40- (xn ) 50@0#0%0%1 #2&( ($.$&$/0,+ 9;-+1 40- x
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n ≥ n0 ⇒ |xn − x| < ǫ
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0} 20&+#0%0% x %$<(2#3 0F0% 40- ,+-+& (242% $&5'='%' 52 40&0,$-'@6 7 M2&5+
|xn − x| < ǫ ⇔ x ∈ (x − ǫ, x + ǫ)
42=3%(3#3 0#(+%+%0 >+-0-6
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1
ǫ]
2
1
an → a =⇒ [(∀ǫ > 0)(∃n0 )(n0 < n ⇒ |an+1 − a| < ǫ]
2
,5;57*'*)*;# J0)595&
an → a =⇒ [(∀ǫ > 0)(∃n0 )(n0 < n ⇒ |an+2 − a| <
n0 < n =⇒ |an+2 − an+1 |
=⇒= |an+2 − a + a − an+1 |
=⇒≤ |an+2 − a| + |a − an+1 |
1
1
=⇒< ǫ + ǫ = ǫ]
2
2
4'+51: -%)%6(*?* 14&0/0 8:65)# @(% ,5&95&B 2%)*'%& ,*&%'-% 75?:&(:1:&95&B
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(an ) -')5#$%$&$ 3+1$'&$*6
an → a ⇐⇒ (∀W ∈ B(a))(∃n0 ) [n0 < n ⇒ an ∈ W ]
⇐⇒ (∀W ∈ B(a))(∃n0 ) [n0 < nk ⇒ ank ∈ W ]
⇐⇒ ank → a
0'216
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$E$*+ 4$1 M0*.&$,0* 0'(-. F%+1+< an = f (n) #$%$&$6 OIA,'+ 4$1 M0*.&$,5
0*2* 3-1'/:/*/ @A&)+1$*$%6P
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-')52%-, )0>0'0?$&$*+ @A1+ n1 → 0 #/16 R ;-'#+ #$%$*$* '$($)$ 3-1#/16
8,1/K- 0 *0.)-&/ #$%$*$* 4$1$K$. ,/:/'(- *0.)-&/#/16
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/ (m − ǫ, m + ǫ)
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an → a =⇒ [(∀ǫ > 0)(∃n0 )(n0 < n ⇒ |an − a| <
1
ǫ]
2
an → a =⇒ [(∀ǫ > 0)(∃n0 )(n0 < m ⇒ |am − a| <
1
ǫ]
2
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n0 < m, n =⇒ |an − am | = |an − a + a − am |
=⇒≤ |an − a| + |a − am |
1
1
=⇒< ǫ + ǫ = ǫ
2
2
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1
an → a ⇐⇒ (∀ǫ > 0)(∃n0 ) n0 < n ⇒ |an − a| < ǫ
2
⇐⇒ (∀ǫ > 0)(∃n0 ) [n0 < n ⇒ an ∈ (a − ǫ, a + ǫ)]
⇐⇒ (∀ǫ > 0)(∃n0 ) [n0 < n ⇒ (an − a) ∈ (−ǫ, +ǫ)]
⇐⇒ (an − a) → 0