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Bolzano-Weierstrass by Bisection
Working with subsequences
MCQ
Which of the following statements about subsequences are true?
Possible answers
If
φ
:
N
→
N
\varphi : \mathbb{N} \to \mathbb{N}
φ
:
N
→
N
is strictly increasing, then
φ
(
n
)
≥
n
\varphi(n) \geq n
φ
(
n
)
≥
n
for all
n
n
n
If two subsequences of
(
u
n
)
(u_n)
(
u
n
)
converge to different limits, then
(
u
n
)
(u_n)
(
u
n
)
diverges
(
u
n
2
)
\left(u_{n^2}\right)
(
u
n
2
)
is not a subsequence of
(
u
n
)
(u_n)
(
u
n
)
, because
n
↦
n
2
n \mapsto n^2
n
↦
n
2
does not reach every integer
If
(
u
2
n
)
\left(u_{2n}\right)
(
u
2
n
)
converges, then
(
u
n
)
(u_n)
(
u
n
)
converges
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