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Vojtěch Jarník IMC
2000 VJIMC
Problem 3
limits equal for bounded sequence of reals
limits equal for bounded sequence of reals
Source: VJIMC 2000 1.3
July 26, 2021
limits
real analysis
Problem Statement
Let
a
1
,
a
2
,
…
a_1,a_2,\ldots
a
1
,
a
2
,
…
be a bounded sequence of reals. Is it true that the fact
lim
N
→
∞
1
N
∑
n
=
1
N
a
n
=
b
and
lim
N
→
∞
1
log
N
∑
n
=
1
N
a
n
n
=
c
\lim_{N\to\infty}\frac1N\sum_{n=1}^Na_n=b\enspace\text{ and }\enspace\lim_{N\to\infty}\frac1{\log N}\sum_{n=1}^N\frac{a_n}n=c
N
→
∞
lim
N
1
n
=
1
∑
N
a
n
=
b
and
N
→
∞
lim
lo
g
N
1
n
=
1
∑
N
n
a
n
=
c
implies
b
=
c
b=c
b
=
c
?
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