Prove that $a \leq 0$ if $a \leq \frac 1 n$ for all $n$

Question

The full question reads: Suppose that $a$ is a number that has the property that for every $n \in \mathbb{N}$, $a \leq 1/n$. Prove $a \leq 0$.

Is there anyway to show this using Archimedean Property, or is it something related to the Completeness Axiom? The problem using the Archimedean Property is that I get up to $a< \epsilon$ but from there I am not able to conclude anything about whether $a \leq 0$ because $\epsilon > 0$.

For any $\varepsilon>0$ exists $n\in\mathbb{N}$ such that $0<\frac{1}{n}<\varepsilon$ — Math.mx, Sep 18 '11 at 04:01
People are really coming out of the woodwork to answer this! — The Chaz 2.0, Sep 18 '11 at 12:50
It is nice that $\mathbb{R}$ is first-countable, isn't it? :) — Mark, Sep 18 '11 at 13:06
"Anyway" is a perfectly respectable word, but it should not be used as a synonym of "any way" (two words). But in the last few years I've seen a number of cases of people doing this. — Michael Hardy, Sep 18 '11 at 14:39

score 8 · Accepted Answer · answered Sep 17 '11 at 23:41

8

In other words, prove that $a\not>0$. Assume $a>0$, what does that tell you about some $n$?

answered Sep 17 '11 at 23:41

ThomasMcLeod

366

score 3 · Answer 2 · answered Sep 17 '11 at 23:41

I'll not give the complete answer to this (on purpose).

But here is some intuition: if you have a positive number, what can you say about that positive number as compared to the numbers $1/n$ for each n?

Start with something concrete, like 0.1 for example. It can't have the property you mention... why not?

The same reason works for all positive numbers.

score 2 · Answer 3 · answered Sep 18 '11 at 01:38

2

If $a \leq 1/n$ for all $n \in \mathbb{N}$, then $a \le \inf \{ 1/n : n \in \mathbb{N} \}=0$. Now, to prove that the inf is $0$, you need the Archimedean Property. (The inf being 0 is actually equivalent to the Archimedean Property.)

answered Sep 18 '11 at 01:38

lhf

216,483

Note also that this uses the Completeness Axiom. In fact, the Completeness Axiom implies the Archimedean Property. – lhf Sep 18 '11 at 15:30

score 1 · Answer 4 · edited Apr 16 '13 at 06:01

1

Suppose that $a>0$. Then by the Archimedean Property, there exists a natural number $n$ so that $1/n<a$. But this is a contradiction. Then $a\leq0$ if $a\leq 1/n$ for all $n$.

edited Apr 16 '13 at 06:01

leo

10,433

answered Sep 18 '11 at 15:11

Roiner Segura Cubero

3,923

score 0 · Answer 5 · answered Sep 18 '11 at 06:40

Assume $a>0$,then there exists some $\epsilon$ such that $0<\epsilon<a$.

Let $N=[\epsilon]+1$,then $\frac{1}{N}<\epsilon<a$, which contradicts to that for every $n∈N, a≤1/n$.

So, $a\leq0$.[Q.E.D]

Above is implied in @Daniel's and @lhf's answers.

score -1 · Answer 6 · edited Sep 18 '11 at 12:38

-1

$n$ is a natural number, therefore it can be $1,2,3, \ldots$. On the other hand, $1/n$ is a continuous function for every real $x$ distinct to zero.

Applying the following limit $$ \mathop {\lim }\limits_{n \to \infty } a \le \mathop {\lim }\limits_{n \to \infty } \frac{1}{n} = 0 $$

$$ a \le 0 $$

$$Q.E.D.$$

edited Sep 18 '11 at 12:38

Srivatsan

26,311

answered Sep 17 '11 at 23:47

Daniel

111

2

$1/n$ is not continuous. – ThomasMcLeod Sep 18 '11 at 01:01
1

Whatever that means. – Did Sep 18 '11 at 08:01
it is continuous for every x distinct to zero, as stated above. – Daniel Sep 18 '11 at 14:28
The problem states $n \in \mathbb{N}$. – ThomasMcLeod Sep 20 '11 at 04:28

Prove that $a \leq 0$ if $a \leq \frac 1 n$ for all $n$

6 Answers6