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Lebl-Basic-Analysis-I
· Problem 2.1.20
Sequences and Series
Mathematics · sequences-and-limits
Problem
Let
{
x
n
}
n
=
1
∞
\{x_n\}_{n=1}^\infty
{
x
n
}
n
=
1
∞
be a sequence and define a sequence
{
y
n
}
n
=
1
∞
\{y_n\}_{n=1}^\infty
{
y
n
}
n
=
1
∞
by
y
2
k
:
=
x
k
2
y_{2k} := x_{k^2}
y
2
k
:=
x
k
2
and
y
2
k
−
1
:
=
x
k
y_{2k-1} := x_k
y
2
k
−
1
:=
x
k
for all
k
∈
N
k \in \mathbb{N}
k
∈
N
. Prove that
{
x
n
}
n
=
1
∞
\{x_n\}_{n=1}^\infty
{
x
n
}
n
=
1
∞
converges if and only if
{
y
n
}
n
=
1
∞
\{y_n\}_{n=1}^\infty
{
y
n
}
n
=
1
∞
converges. Furthermore, prove that if they converge, then
lim
n
→
∞
x
n
=
lim
n
→
∞
y
n
\lim_{n\to\infty} x_n = \lim_{n\to\infty} y_n
lim
n
→
∞
x
n
=
lim
n
→
∞
y
n
.
Related theory
sequences-and-limits
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