Theory check

Calculus Level 4

Which of these statements are always true?

A: If $\{ a_n \}$ is Cauchy sequence, then the sequence converges in any metric space
B: If $\{ b_n \}$ bounded sequence, then it must have a sub-sequence that converges
C: If $\{ c_n \}$ is strictly monotonic sequence, then it converges to its infimum or supremum
D: If $\{ d_n \}$ is sequence that converges, every sub-sequence of $\{ d_n \}$ converges as well
E: If sequences $\{ x_n \}$ and $\{ y_n \}$ , satisfy for each $n$ : $x_n > y_n$ . Then, the following will be satisfied as well: $\ \lim_{x\to\infty } x_n > \lim _{ y \rightarrow \infty }{ y_n }$

Only

B

D

and

E

Only

E

B

and

D

B

C

and

D

All of them

A

and

E

C

D

and

E

1 solution

Milan Milanic
Nov 8, 2016

Solution:

It is very important to understand that these sequences have infinite number of elements.

A: false

A1. Any sequence that converges is automatically Cauchy's. Reverse is not always true.

A2. If sequence is Cauchy's, then it converges iff it is in complete metric space . (Basically, the definition for complete metric spaces is: a metric space is complete iff every Cauchy's sequence converges).

B: true

B1. Bolzano-Weierstrass theorem

C: false

C1. Example: $c_n = n$ , for every n . This sequence is strictly monotonic (strictly increasing), yet it diverges.

D: true

D1.

$(\forall R > 0)(\exists n_0 \in N)(\forall n \in N)\ ((n \ge n_0)\Rightarrow (d_n \in L(d, R)))$ [1]

For any infinite $M \subset N$ where $m \in M$ , sequence $\{d_m\}$ will be sub-sequence of $\{ d_n \}$ , so that:

$(\exists m_0 \in M)(\forall m_n \in M)\ ((m_n \ge m_0 \ge n_0) \Rightarrow (d_{m_{n}} \in L(d, R)))$ [2]

[1] $\{d_n\}$ converges to $d$

[2] Any sub-sequence of $\{d_n\}$ converges to $d$ as well.

E: false

E1. Example: $x_n = \frac{2}{n}$ and $y_n = \frac{1}{n}$ . Clearly $x_n > y_n$ for all n . However, both sequences converge to $0$ .

E2. If last " $\gt$ " is replaced with " $\ge$ " from statement E, a very similar statement will be made, but this one is always true.

Conclusion: Statements $B$ and $D$ are always true.

For A, it is better to make it clear that "Whether a sequence is Cauchy depends on the metric that is being used. Just because one metric makes it Cauchy, doesn't mean that it will be Cauchy in all metrics".

Calvin Lin Staff - 4 years, 7 months ago

For A I would also include a specific counter-example.

Jason Dyer Staff - 4 years, 7 months ago

In statement B, if we consider a bounded sequence on a non-complete metric space, will it also converge? The Bolzano–Weierstrass theorem only regards sequences in the Euclidian space R^n.

Lucas Viana Reis - 2 years, 6 months ago

Theory check

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