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Vojtěch Jarník IMC
2010 VJIMC
Problem 2
Problem 2
Part of
2010 VJIMC
Problems
(2)
A,B matrix
Source:
4/4/2013
If
A
,
B
∈
M
2
(
C
)
A,B\in M_2(C)
A
,
B
∈
M
2
(
C
)
such that
A
B
−
B
A
=
B
2
AB-BA=B^2
A
B
−
B
A
=
B
2
then prove that
A
B
=
B
A
AB=BA
A
B
=
B
A
linear algebra
matrix
linear algebra unsolved
a_(n-1)-a_n->0 and a_(2n)-2a_n->0
Source: VJIMC 2010 2.2
6/4/2021
Prove or disprove that if a real sequence
(
a
n
)
(a_n)
(
a
n
)
satisfies
a
n
+
1
−
a
n
→
0
a_{n+1}-a_n\to0
a
n
+
1
−
a
n
→
0
and
a
2
n
−
2
a
n
→
0
a_{2n}-2a_n\to0
a
2
n
−
2
a
n
→
0
as
n
→
∞
n\to\infty
n
→
∞
, then
a
n
→
0
a_n\to0
a
n
→
0
.
Sequences
limits