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Sweden Contests
Swedish Mathematical Competition
1966 Swedish Mathematical Competition
1
lim {x_n + - y_n} = 0 ? id lim {x_n} = lim {y_n} = 0
lim {x_n + - y_n} = 0 ? id lim {x_n} = lim {y_n} = 0
Source: 1966 Swedish Mathematical Competition p1
March 21, 2021
limit
algebra
Sequence
Problem Statement
Let
{
x
}
\{x\}
{
x
}
denote the fractional part of
x
x
x
,
x
−
[
x
]
x - [x]
x
−
[
x
]
. The sequences
x
1
,
x
2
,
x
3
,
.
.
.
x_1, x_2, x_3, ...
x
1
,
x
2
,
x
3
,
...
and
y
1
,
y
2
,
y
3
,
.
.
.
y_1, y_2, y_3, ...
y
1
,
y
2
,
y
3
,
...
are such that
lim
{
x
n
}
=
lim
{
y
n
}
=
0
\lim \{x_n\} = \lim \{y_n\} = 0
lim
{
x
n
}
=
lim
{
y
n
}
=
0
. Is it true that
lim
{
x
n
+
y
n
}
=
0
\lim \{x_n + y_n\} = 0
lim
{
x
n
+
y
n
}
=
0
?
lim
{
x
n
−
y
n
}
=
0
\lim \{x_n - y_n\} = 0
lim
{
x
n
−
y
n
}
=
0
?
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