1回: https://www.nicovideo.jp/watch/sm34083025 https://www.youtube.com/watch?v=hhv7WDGPJb0&t=19s 2回: https://www.nicovideo.jp/watch/sm34151019 https://www.youtube.com/watch?v=BF7g2EevsEg 3回: https://www.nicovideo.jp/watch/sm34185874 https://youtu.be/oUMS5Z7Nj70 4回: https://www.nicovideo.jp/watch/sm34195873 https://youtu.be/WUJAAxF8vRA 5回: https://www.nicovideo.jp/watch/sm34271556 https://youtu.be/EYcGCuWVpdU

2. “No one should be ashamed to admit they are wrong,
which is but saying, in other words,
that they are wiser today than they were yesterday”
— Alexander Pope —

3. G = (N, {Qi}i∈N, {Fi}i∈N)

8.
Rn a = (a1, a2, ⋯, an)
a, b
{λa + μb|0 ≤ λ ≤ 1, 0 ≤ μ ≤ 1, λ + μ = 1}
Rn
a, b
Rn a, b, c
{λa + μb + νc|0 ≤ λ ≤ 1, 0 ≤ μ ≤ 1, 0 ≤ ν ≤ 1, λ + μ + ν = 1}
a, b, c
Rn a0
, a1
, a2
, ⋯, ar
{
r
∑
i=0
λi
ai
|0 ≤ λi
≤ 1,
r
∑
i=0
λi
= 1}
a1
− a0
, a2
− a0
, ⋯, ar
− a0
a
b
c

9.
(1,0), (0,1)
{λ(1,0) + μ(0,1)|0 ≤ λ ≤ 1, 0 ≤ μ ≤ 1, λ + μ = 1}
R2
Rm (1,0,0,⋯,0), (0,1,0,⋯,0), ⋯, (0,0,0,⋯,1)
{
m
∑
i=1
λi
ai
|0 ≤ λi
≤ 1,
m
∑
i=1
λi
= 1}
= {(λ, μ)|0 ≤ λ ≤ 1, 0 ≤ μ ≤ 1, λ + μ = 1}
= {(λ,1 − λ)|0 ≤ λ ≤ 1}
R3 (1,0,0), (0,1,0), (0,0,1)

10.
a1
, a2
, ⋯, ak
∈ Rn
∀λ1, λ2, ⋯, λk ∈ R, λ1a1
+ λ2a2
+ ⋯ + λkak
= 0 ⇒ λ1 = λ2 = ⋯ = λk = 0
Rn
a
b
c
a, b, c
b − a, c − a
a, b, c a = b or c = t(b − a) + a
λ1(b − a) + λ2(c − a) = λ2(c − a)
a = b
λ2 = 0 λ1
b ≠ a, c = t(b − a) + a
λ1(b − a) + λ2(c − a) = λ1(b − a) + λ2t(b − a) = (λ1 + λ2t)(b − a)
λ1 = − tλ2 λ1, λ2

11.
a0
, a1
, a2
, ⋯, ak
∈ Rn
∀λ1, λ2, ⋯, λk ∈ R, λ1a1
+ λ2a2
+ ⋯ + λkak
= 0 ⇒ λ1 = λ2 = ⋯ = λk = 0
Rn
a
b
c
a, b, c
b − a, c − a
λ1(b − a) + λ2(c − a) = 0
b − a, c − a
(λ1, λ2) ≠ (0, 0)
λ2 = 0 (λ1 ≠ 0) λ1(b − a) = 0 ⇔ b = a
λ2 ≠ 0, λ1 ≠ 0
λ1(b − a) + λ2(c − a) = 0 ⇔ c =
λ1
λ2
(b − a) + a

12.
a0
, a1
, a2
, ⋯, ak
∈ Rn
∀λ1, λ2, ⋯, λk ∈ R, λ1a1
+ λ2a2
+ ⋯ + λkak
= 0 ⇒ λ1 = λ2 = ⋯ = λk = 0
Rn a, b, c
b − a, c − a
b − a, c − a
a, b, c
Rn a0
, a1
, a2
, ⋯, ar
{
r
∑
i=0
λi
ai
|0 ≤ λi
≤ 1,
r
∑
i=0
λi
= 1} a1
− a0
, a2
− a0
, ⋯, ar
− a0

15.
d(x1, x2) =
n
∑
i=1
(xi
1 − xi
2)2
x1 = (x1
1, x2
1, ⋯, xn
1) x2 = (x1
2, x2
2, ⋯, xn
2)
d((0,1), (2,2)) = (0 − 2)2
+ (1 − 2)2
= 5
x1
x2

16.
d(x, a) < r
Q ⊂ Rn
a = (a1
, a2
, ⋯, an
)
x1
x2
x = (x1
, x2
, ⋯, xn
) ∈ Q
r = 4, a = (0, 0)r = 4

17.
x ∈ Q
Q ⊂ Rn x1, x2, ⋯
lim
n→∞
xn = x
(1 −
1
n
,
1
n
) → (1 ,0) ∈ Q (1 −
1
n
,
1
n
) → (1 ,0) ∉ Q

18.
Q ⊂ Rn
tx1 + (1 − t)x2 ∈ Q
tx1 + (1 − t)x2 = (tx1
1 + (1 − t)x1
2, tx2
1 + (1 − t)x2
2) ∈ Q
x1 x2 t ∈ [0, 1]
x1
x2
x1
x2
x1
x2
tx1 + (1 − t)x2 ∉ Q

21.
x xx x xx

22.
10
Fi(qi, q−i) Qi
Fi(qi, q−i)
qi
10
Fi(qi, q−i)
qi

24.
F : X → Y x ∈ X F(x) ⊂ Y
X = {1,2,3}, Y = {5,6,7}
F(1) = {5}, F(2) = {5,7}, F(3) = ϕ
X = [0, 1], Y = [0, 1]
F(x) =
1 x < 1/2
[0, 1] x = 1/2
0 x > 1/2
F(x) F(x)

25.
F : X → Y x ∈ X F(x) ⊂ Y
F* : X → 2Y
Y = {5,6,7} → 2Y
= {ϕ, {5}, {6}, {7}, {5,6}, {5,7}, {6,7}, {5,6,7}}
X = {1,2,3}
F*(1) = {5}, F*(2) = {5,7}, F*(3) = ϕ

26.
F : X → Y x ∈ X F(x) ⊂ Y
F : X → Y
x ∈ X F(x) ⊆ Y F
x ∈ X F(x) ⊆ Y F
x ∈ X F(x) ⊆ Y F
x ∈ X F(x) ⊆ Y F
x ∈ X F(x) ⊆ Y F
F(x) =
1 x < 1/2
[0, 1] x = 1/2
0 x > 1/2
F(x) =
1 x < 1/2
[0, 1) x = 1/2
0 x > 1/2

28.
F : X → Y x F(x) ⊆ V V ⊆ Y
x O ⊆ X x′ ∈ O F(x′) ⊆ V
F : X → Y x F(x) ∩ V ≠ ϕ V ⊆ Y
x O ⊆ X x′ ∈ O F(x′) ∩ V ≠ ϕ
x ∈ X
F : X → Y

29.
F : X → Y x F(x) ⊆ V V ⊆ Y
x O ⊆ X x′ ∈ O F(x′) ⊆ V
x
F(x)
x
V
O
x′ x′
F(x′) F(x′)
O
F(x′) ⊈ V
V

30.
F : X → Y x
x O ⊆ X x′ ∈ O
F(x) ∩ V ≠ ϕ V ⊆ Y
F(x′) ∩ V ≠ ϕ
x
F(x)
x
F(x)
V
O
F(x′)
x′
F(x′) ∩ V ≠ ϕ
O
V
x′
F(x) ∩ V ≠ ϕ
F(x′) ∩ V ≠ ϕ

31.
x1
y1 ∈ F(x1)
xx2 xv
F(x)
yv ∈ F(xv)
y ∈ F(x)
y2 ∈ F(x2)
{xv}∞
v=1, {yv}∞
v=1
yv ∈ F(xv), v = 1,2,… xv → x, yv → y (v → ∞)
y ∈ F(x)
x1
y1 ∈ F(x1)
x x2xv
F(x)
yv ∈ F(xv)
y ∉ F(x)
y2 ∈ F(x2)
yν = y
F : X → Y
xν = x + 1/ν
x

32.
x1
y1 ∈ F(x1)
xx2 xv
F(x)
yv ∈ F(xv)
y ∈ F(x)
y2 ∈ F(x2)
x1
y1 ∈ F(x1)
x x2xv
F(x)
yv ∈ F(xv)
y ∉ F(x)
y2 ∈ F(x2)
yν = y
F : X → Y
xν = x + 1/ν
{(x, y) ∈ X × Y|y ∈ F(x)}

33.
{xv}∞
v=1, xv → x (v → ∞)
y ∈ F(x) yv → y (v → ∞) yv ∈ F(xv)
x1
y1 ∈ F(x1)
x x2xv
F(x)
yv ∈ F(xv)
y ∈ F(x)
y2 ∈ F(x2)
yν = y
F : X → Y
xν = x + 1/ν
{yv}∞
v=1
x1
y1 ∈ F(x1)
xx2 xv
F(x)
yv ∈ F(xv)
y ∈ F(x)
y2 ∈ F(x2)
yν = y′( ≠ y)
xν = x − 1/ν
x

34.
G = (N, {Qi}i∈N, {Fi}i∈N)
N
Fi
Qi
q−i = (q1, ⋯, qi−1, qi+1, ⋯, qn)
Bi(q−i) = {qi ∈ Qi |Fi(qi, q−i) = max
ri∈Qi
Fi(ri, q−i)}
Bi : Q1 × … × Qi−1 × Qi+1 × … × Qn → Qi

35.
BA(q−A) = {qA ∈ QA |FA(qA, q−A) = max
rA∈QA
FA(rA, q−A)}
A B
⟺ BA(qB) = {qA ∈ QA |FA(qA, qB) = max
rA∈QA
FA(rA, qB)}
qA = (qA1, qA2)
qA1
qA2
qA2
qA1
QAFA(rA, qB) = 1rA1qB1 + (−1)rA2qB1 + (−1)rA1qB2 + 1rA2qB2
= 1rA1qB1 + (−1)(1 − rA1)qB1 + (−1)rA1(1 − qB1) + 1(1 − rA1)(1 − qB1)
= 4rA1qB1 − 2rA1 − 2qB1 + 1
= 2(2qB1 − 1)rA1 − 2qB1 + 1

36.
A B
qA1
qA2
qA2
qA1
QA
BA(qB) = {qA ∈ QA |FA(qA, qB) = max
rA∈QA
(2(2qB1 − 1)rA1 − 2qB1 + 1)}
qB1
qA1
BA(qB) =
8
<
:
(0, 1) (0  qB1 < 1/2)
8qA 2 QA (qB1 = 1/2)
(1, 0) (1/2 < qB1  1)<latexit sha1_base64="/DP8a7mV5I7gdP6QXNjyEY55Kh4=">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</latexit><latexit sha1_base64="/DP8a7mV5I7gdP6QXNjyEY55Kh4=">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</latexit><latexit sha1_base64="/DP8a7mV5I7gdP6QXNjyEY55Kh4=">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</latexit><latexit sha1_base64="/DP8a7mV5I7gdP6QXNjyEY55Kh4=">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</latexit><latexit sha1_base64="/DP8a7mV5I7gdP6QXNjyEY55Kh4=">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</latexit>
qA2
qA1
0 ≤ qB1 < 1/2
qB1 = 1/2
1/2 < qB1 ≤ 1

37.
q = (q1, ⋯, qn)
q * ∈ B(q*)
B(q) = B1(q−1) × ⋯ × Bn(q−n)
q*
F(q*) ≥ F(qi, q*−i
) ∀qi ∈ Qi
q*
B : Q1 × … × Qn → Q1 × … × Qn

38.
BA(qB) = {qA ∈ QA |FA(qA, qB) = max
rA∈QA
FA(rA, qB)}
BB(qA) = {qB ∈ QB |FB(qB, qA) = max
rB∈QB
FB(rB, qA)}
B(qA, qB) = BA(qB) × BB(qA)
= {(qA, qB) ∈ QA × QB |FA(qA, qB) = max
rA∈QA
FA(rA, qB), FB(qB, qA) = max
rB∈QB
FB(rB, qA)}
B((1,0), (0,1)) = BA((0,1)) × BB((1,0)) = {((0,1), (1,0))}
⟺ BA(qB) =
(0, 1) 0 ≤ qB1 < 1/2
∀qA ∈ QA qB1 = 1/2
(0, 1) 0 < qB1 ≤ 1
⟺ BB(qA) =
(1, 0) 0 ≤ qA1 < 1/2
∀qB ∈ QB qA1 = 1/2
(1, 0) 0 < qA1 ≤ 1
qA = (qA1, qA2)
qA = (1,0), qB = (0,1)
= {(qA, qB) ∈ BA(qB) × BB(qA)}
qB1 = 0 qA1 = 1

39.
BA(qB) = {qA ∈ QA |FA(qA, qB) = max
rA∈QA
FA(rA, qB)}
BB(qA) = {qB ∈ QB |FB(qB, qA) = max
rB∈QB
FB(rB, qA)}
B(qA, qB) = BA(qB) × BB(qA)
= {(qA, qB) ∈ QA × QB |FA(qA, qB) = max
rA∈QA
FA(rA, qB), FB(qB, qA) = max
rB∈QB
FB(rB, qA)}
B((1/2,1/2), (1/2,1/2)) = BA((1/2,1/2)) × BB((1/2,1/2)) = {(qA, qB)}
⟺ BA(qB) =
(0, 1) 0 ≤ qB1 < 1/2
∀qA ∈ QA qB1 = 1/2
(0, 1) 0 < qB1 ≤ 1
⟺ BB(qA) =
(1, 0) 0 ≤ qA1 < 1/2
∀qB ∈ QB qA1 = 1/2
(1, 0) 0 < qA1 ≤ 1
qA = (qA1, qA2)
qA = (1/2,1/2), qB = (1/2,1/2)
= {(qA, qB) ∈ BA(qB) × BB(qA)}
qB1 = 1/2 qA1 = 1/2 (1/2,1/2) ∈ {(∀qA, ∀qB)}

42.
q = (q1, ⋯, qn)
q * ∈ B(q*)
B(q) = B1(q−1) × ⋯ × Bn(q−n)
q*
F(q*) ≥ F(qi, q*−i
) ∀qi ∈ Qi
q*
Bi(q−i) = {qi ∈ Qi |Fi(qi, q−i) = max
ri∈Qi
Fi(ri, q−i)}
= {qi ∈ Qi |Fi(qi, q−i) ≥ Fi(ri, q−i) ∀ri ∈ Qi}

43.
x = f(x)
q * ∈ B(q*)
x ∈ F(x)
f(x) F(x)

44.
x * ∈ H(x*)
x ∈ S H(x)
x *
H
H
H

45.
G = (N, {Qi}i∈N, {Fi}i∈N)
B : Q → Q
Rm
Qi
Q = Q1 × ⋯ × Qn

46.
G = (N, {Qi}i∈N, {Fi}i∈N)
q ∈ Q B(q) Q
Bi(q−i)
Bi(q−i) = {qi ∈ Qi |Fi(qi, q−i) = max
ri∈Qi
Fi(ri, q−i)}
q−i ∈ Q−i
Fi(qi, q−i) qi
q1
i , q2
i ∈ Bi(q−i)
tq1
i + (1 − t)q2
i ∈ Bi(q−i), ∀t ∈ [0,1]

47.
10
Fi(qi, q−i)
qi
10
Fi(qi, q−i)
qi
q1
i , q2
i ∈ Bi(q−i) ⇒
tq1
i + (1 − t)q2
i ∈ Bi(q−i), ∀t ∈ [0,1]

48.
10
Fi(qi, q−i)
qi
10
Fi(qi, q−i)
qi
q1
i q2
i
q1
i , q2
i ∈ Bi(q−i) ⇒
tq1
i + (1 − t)q2
i ∈ Bi(q−i), ∀t ∈ [0,1]

49.
G = (N, {Qi}i∈N, {Fi}i∈N)
q ∈ Q B(q) Q
(1) Fi(q1
i , q−i) ≥ Fi(ri, q−i) ∀ri ∈ Qi
q1
i , q2
i ∈ Bi(q−i)
⇒ tFi(q1
i , q−i) + (1 − t)Fi(q2
i , q−i) ≥ Fi(ri, q−i) ∀ri ∈ Qi
(2) Fi(q2
i , q−i) ≥ Fi(ri, q−i) ∀ri ∈ Qi
⇔ Fi(tq1
i , tq−i) + Fi((1 − t)q2
i , (1 − t)q−i) ≥ Fi(ri, q−i) ∀ri ∈ Qi
⇔ Fi(tq1
i + (1 − t)q2
i , q−i) ≥ Fi(ri, q−i) ∀ri ∈ Qi
tq1
i + (1 − t)q2
i ∈ Bi(q−i)

50.
G = (N, {Qi}i∈N, {Fi}i∈N)
q ∈ Q B(q) Q
Bi(q−i)
Bi(q−i) = {qi ∈ Qi |Fi(qi, q−i) = max
ri∈Qi
Fi(ri, q−i)}
q−i ∈ Q−i
Fi(qi, q−i) qi
q1
i , q2
i ∈ Bi(q−i)
tq1
i + (1 − t)q2
i ∈ Bi(q−i), ∀t ∈ [0,1]

51.
G = (N, {Qi}i∈N, {Fi}i∈N)
q ∈ Q B(q) Q
B(q) = B1(q−1) × ⋯ × Bn(q−n)
q = (q1, ⋯, qn) ∈ Q
Q
q1
, q2
∈ B(q)
tq1
+ (1 − t)q2
= t(q1
1, ⋯, q1
n) + (1 − t)(q2
1, ⋯, q2
n) = (tq1
1 + (1 − t)q2
1, ⋯, tq1
n + (1 − t)q2
n)
tq1
k + (1 − t)q2
k ∈ Bk(q−k)
t ∈ [0,1]
tq1
+ (1 − t)q2
∈ B(q)

52.
G = (N, {Qi}i∈N, {Fi}i∈N)
B(q)
{qv
}∞
v=1Q = Q1 × ⋯ × Qn
qv
→ q, qv
→ q (v → ∞)
q ∈ B(q)
{qv
}∞
v=1
qv
∈ B(qv
)

53.
G = (N, {Qi}i∈N, {Fi}i∈N)
B(q)
{qv
}∞
v=1Q = Q1 × ⋯ × Qn
qv
→ q, qv
→ q (v → ∞)
{qv
}∞
v=1
qv
∈ B(qv
)
Fi(qv
i , qv
−i) ≥ Fi(ri, qv
−i) v = 1,2,⋯
i ri ∈ Qi
Fi(qi, q−i) ≥ Fi(ri, q−i)
v → ∞ Fi
qi ∈ Bi(q−i)
lim
v→∞
Fi(qv
i , qv
−i) = Fi( lim
v→∞
qv
i , lim
v→∞
qv
−i) = Fi(qi, q−i)

54.
G = (N, {Qi}i∈N, {Fi}i∈N)
B(q)
Fi(qi, q−i) < Fi(ri, q−i)
|Fi(ri, qv′
−i) − Fi(ri, q−i)| < ϵ
v′
|Fi(qv′
i , qv′
−i) − Fi(qi, q−i)| < ϵ lim
v→∞
Fi(qv
i , qv
−i) = Fi(qi, q−i)
lim
v→∞
Fi(ri, qv
−i) = Fi(ri, q−i)
ϵ
Fi(qi, q−i) + α = Fi(ri, q−i)
Fi(qv
i , qv
−i) ≥ Fi(ri, qv
−i) v = 1,2,⋯
Fi(ri, qv′
−i) > Fi(ri, q−i) − ϵ = Fi(qi, q−i) + α − ϵ > Fi(qv′
i , qv′
−i) + α − 2ϵ = Fi(qv′
i , qv′
−i)
ϵ = α/2
Fi(qi, q−i) ≥ Fi(ri, q−i)

55.
G = (N, {Qi}i∈N, {Fi}i∈N)
B(q)
qi ∈ Bi(q−i)
q = (q1, ⋯, qn) ∈ B(q−1) × ⋯ × B(q−n) = B(q)
i

56.
x * ∈ H(x*)
x ∈ S H(x)
x *
H
H
H
q * ∈ B(q*)