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Lebl-Basic-Analysis-I
· Problem 2.4.4
Sequences and Series
Mathematics · cauchy-sequences
Problem
Let
{
x
n
}
n
=
1
∞
\{x_n\}_{n=1}^\infty
{
x
n
}
n
=
1
∞
and
{
y
n
}
n
=
1
∞
\{y_n\}_{n=1}^\infty
{
y
n
}
n
=
1
∞
be sequences such that
lim
n
→
∞
y
n
=
0
\lim_{n\to\infty} y_n = 0
lim
n
→
∞
y
n
=
0
. Suppose that for all
k
∈
N
k \in \mathbb{N}
k
∈
N
and for all
m
≥
k
m \geq k
m
≥
k
, we have
∣
x
m
−
x
k
∣
≤
y
k
.
|x_m - x_k| \leq y_k.
∣
x
m
−
x
k
∣
≤
y
k
.
Show that
{
x
n
}
n
=
1
∞
\{x_n\}_{n=1}^\infty
{
x
n
}
n
=
1
∞
is Cauchy.
Related theory
Sequences Convergence Cauchy Sequences Subsequences
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