0

I know a similar question has been posted at Negating the Definition of a Convergent Sequence to Find the Definition of a Divergent Sequence. But there is one thing I don't really understand, or I should say there is one thing I don't really agree with in one of the solutions to this post.

Let's state the definition of convergence of a sequence first: "$\forall\varepsilon>0$ $\exists N\in\Bbb R$ $\forall n\in\Bbb R, n > N $such that $|x_n -\mathscr l|<\varepsilon$ $\forall n \in \Bbb N$.

Well I think the negation of the above statement should be: $$\exists\ \epsilon > 0,\ \forall\ N \in \mathbb N\ \exists\ \mathbb N \ni n \leq N : |x_n - l| \ge \epsilon$$

But it seems like everyone in that post agree that:

$$ \exists\ \epsilon > 0,\ \forall\ N \in \mathbb N\ \exists\ \mathbb N \ni n > N : |x_n - l| \ge \epsilon $$

is the correct negation.

Is there somewhere I made a mistake? In my symbolic logic course my professor taught us both the quantifiers and also the statement attached to that quantifiers should be negated together.

I feel confused...

Metaozis
  • 177

2 Answers2

1

The negation of 'for all $n >N$ $P_n$ is true ' is 'there exists $n >N$ such that $P_n$ is false'. It is not 'there exists $n \leq N$ such that $P_n$ is false'.

Kavi Rama Murthy
  • 311,013
  • You are saying $\forall n, n \gt N$ is true is implying $\exists n, not(n \gt N)$ is true, but isn't this essentially $\exists n, n \le N$ is true? – Metaozis Jan 22 '20 at 23:43
  • No, you are not supposed to to change $n>N$ to $n \leq N$. @Metaozis – Kavi Rama Murthy Jan 22 '20 at 23:45
  • Why is so? I know quantifiers are used to indicate the range, or set, that the element will come from, isn't $\forall n, n \gt N$ just equivalent to $\forall n \in \Bbb R, n \gt N$ ? Since $n \gt N$ is essentially the statement that will determine whether is true of false? – Metaozis Jan 22 '20 at 23:48
0

Your definition has $\exists\ \epsilon > 0,\ \forall\ N \in \mathbb N\ \exists\ \mathbb N \ni n \leq N : |x_n - l| \ge \epsilon $.

The mistake is in your use of $n \leq N $ and $\forall\ N \in \mathbb N\ $.

Your definition says that if any initial segment of the $x_i$ differ from $l$ by $\epsilon$, then the series diverges. This would imply that almost all sequences, especially monotonic ones, diverge.

What is needed for divergence is that, for some $\epsilon > 0$, no matter how far out you go, there is an element that differs from $l$ by at least $\epsilon$.

marty cohen
  • 107,799