Prove $(y_n)$ Converges to a Real Number Given $|y_{n+1}-y_n| \leq 2^{-n}$

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Discussion Overview

The discussion revolves around proving that the sequence $(y_n)$ converges to a real number given the condition $|y_{n+1}-y_n| \leq 2^{-n}$. Participants explore whether this condition implies that $(y_n)$ is a Cauchy sequence and what further steps are necessary to establish convergence.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • Some participants suggest that the condition $|y_{n+1}-y_n| \leq 2^{-n}$ implies that $(y_n)$ is a Cauchy sequence.
  • Others argue that it has not yet been established that $(y_n)$ is a Cauchy sequence, raising concerns about the implications of the given condition.
  • A participant proposes a method to show that $(y_n)$ is Cauchy by using the triangle inequality and summing the differences, leading to a bound involving $2^{-n}$.
  • There is a challenge regarding the validity of a specific inequality used in the proof, with participants questioning whether $(n-m+2)2^{-n} \leq 2^{-n+1}$ holds true.
  • Participants discuss the correct formulation of the bounds and the implications of the indices $n$ and $m$ in the context of the inequalities.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether the condition implies that $(y_n)$ is a Cauchy sequence. There are competing views on the validity of certain inequalities and the steps needed to prove convergence.

Contextual Notes

Some assumptions about the indices and the nature of the inequalities remain unresolved, particularly regarding the bounds used in the proof of Cauchy sequence properties.

evinda
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Hello! (Wave)

Let $(y_n)$ be a sequence of numbers such that $|y_{n+1}-y_n| \leq 2^{-n}$ for each $n \in \mathbb{N}$.
Show that the sequence $(y_n)$ converges to a real number.

Doesn't $|y_{n+1}-y_n| \leq 2^{-n}$ for each $n \in \mathbb{N}$ imply that $(y_n)$ is a Cauchy sequence?

So does it remain to show that every Cauchy sequence $(y_n)$ converges to a real number? If so how can we show this? (Thinking)
 
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evinda said:
Hello! (Wave)

Let $(y_n)$ be a sequence of numbers such that $|y_{n+1}-y_n| \leq 2^{-n}$ for each $n \in \mathbb{N}$.
Show that the sequence $(y_n)$ converges to a real number.

Doesn't $|y_{n+1}-y_n| \leq 2^{-n}$ for each $n \in \mathbb{N}$ imply that $(y_n)$ is a Cauchy sequence?

So does it remain to show that every Cauchy sequence $(y_n)$ converges to a real number? If so how can we show this? (Thinking)

Hey evinda! (Smile)

If it's a Cauchy sequence, there's nothing more to do - it will converge to some real number.
However... it's not implied yet that it's a Cauchy sequence. (Worried)
 
I like Serena said:
Hey evinda! (Smile)

If it's a Cauchy sequence, there's nothing more to do - it will converge to some real number.
However... it's not implied yet that it's a Cauchy sequence. (Worried)

A ok... I have thought the following:

We fix a $n \in \mathbb{N}$. We choose a $m \geq n+1$. Then

$|y_m-y_n|=|y_m-y_{m-1}+y_{m-1}+ \dots+ y_{n+1}-y_n| \overset{\text{ Triangle inequality}}{\leq} |y_m-y_{m-1}|+ \dots+ |y_{n+1}-y_n| \leq 2^{-(m-1)}+ \dots+ 2^{-n} \leq (n-m+2) 2^{-n}\leq 2^{-n+1}$.

This holds for any $n \in \mathbb{N}$ so the sequence is Cauchy.
Am I right? (Thinking)
 
evinda said:
$(n-m+2) 2^{-n}\leq 2^{-n+1}$.

I think this doesn't hold. (Worried)
 
I like Serena said:
I think this doesn't hold. (Worried)

Oh yes, right. It holds that $n \leq m-1$ and so $(n-m+2)2^{-n} \leq 2^{-n}$. Right?
 
evinda said:
$|y_m-y_{m-1}|+ \dots+ |y_{n+1}-y_n| \leq 2^{-(m-1)}+ \dots+ 2^{-n} \leq (n-m+2) 2^{-n}\leq 2^{-n+1}$.

This holds for any $n \in \mathbb{N}$ so the sequence is Cauchy.
Am I right? (Thinking)

evinda said:
Oh yes, right. It holds that $n \leq m-1$ and so $(n-m+2)2^{-n} \leq 2^{-n}$. Right?

Hold on! (Wait)
Shouldn't it be:
$$|y_m-y_{m-1}|+ \dots+ |y_{n+1}-y_n| \leq 2^{-(m-1)}+ \dots+ 2^{-n} \leq (m-n) 2^{-n}$$
? (Wondering)
 
I like Serena said:
Hold on! (Wait)
Shouldn't it be:
$$|y_m-y_{m-1}|+ \dots+ |y_{n+1}-y_n| \leq 2^{-(m-1)}+ \dots+ 2^{-n} \leq (m-n) 2^{-n}$$
? (Wondering)
Why isn't it right that

$$ 2^{-(m-1)}+ \dots+ 2^{-n} \leq (m-n) 2^{-n} \leq (n-(m-1)+1) 2^{-n}$$

? (Thinking)
 

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