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IMC
1996 IMC
7
7
Part of
1996 IMC
Problems
(1)
IMC 1996 Problem 7
Source: IMC 1996
3/4/2021
Prove that if
f
:
[
0
,
1
]
→
[
0
,
1
]
f:[0,1]\rightarrow[0,1]
f
:
[
0
,
1
]
→
[
0
,
1
]
is a continuous function, then the sequence of iterates
x
n
+
1
=
f
(
x
n
)
x_{n+1}=f(x_{n})
x
n
+
1
=
f
(
x
n
)
converges if and only if
lim
n
→
∞
(
x
n
+
1
−
x
n
)
=
0
\lim_{n\to \infty}(x_{n+1}-x_{n})=0
n
→
∞
lim
(
x
n
+
1
−
x
n
)
=
0
function
real analysis
Convergence