Distinct reals
x
,
y
in
X
ε
λ
must lie at least 2ε apart:
|
x
-
y
|
>
2ε.
For if instead
0 < |
x
–
y
| < 2ε ,
then
f
(
V
ε
(
x
) ∩
V
ε
(
y
)) ⊂
V
ε
λ
(
x
) ∩
V
ε
λ
(
y
) = ∅,
while
V
ε
(
x
) ∩
V
ε
(
y
) = {((
x
+
y
)/2, |
x
-
y
|/2)} ≠ ∅.
[ >> ]
Distinct reals x, y in
Xελ must lie at least 2ε
apart: |x - y| > 2ε.
For if instead
0 < |x – y| < 2ε ,
then
f(Vε(x) ∩
Vε(y)) ⊂
Vελ(x) ∩
Vελ(y) =
∅,
while
Vε(x) ∩
Vε(y) =
{((x+y)/2, |x-y|/2)}
≠ ∅.