For every natural number $x$, $\frac{1}{x} \le 1$

Question

I am quite new to proofs, and I tried solving this proof myself. Here is what I got:

let $x$ be a natural number
Suppose $\frac{1}{x} > 1$, then $1 > x $ This is a contradiction because x cannot equal $0$ and $x$ must be greater than $0$.

Therefore, For every natural number $x$, $\frac{1}{x}\le 1$

I do not know if this valid, But I tried using proof by contradiction. Is this the correct way of doing it? or am I wrong.

Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. — José Carlos Santos, Oct 11 '19 at 07:31
One thing to be careful of: In some conventions, $0$ is a natural number. — Barry Cipra, Oct 11 '19 at 07:39

score 1 · Accepted Answer · answered Oct 11 '19 at 07:35

1

If $1/x>1$, then multiplication with natural number $x\geq 1$ (multiplication with $x>0$ is monotonous) yields $1>x$ which is a contradiction. Thus $1/x>1$ is false, i.e., $1/x\leq 1$ is true.

answered Oct 11 '19 at 07:35

Wuestenfux

20,964

1

Thanks for the verification! Proofs are really confusing me and I wasn't sure if what I did was correct or not. – Jr194 Oct 11 '19 at 07:41

score 0 · Answer 2 · edited Oct 13 '19 at 14:35

0

You are right Because you cannot change sides of the terms in inequalities , like equations when they divide each other.

$$\frac1x<1 \implies 1\lt x$$ (This is not applicable for the inequalities.)

edited Oct 13 '19 at 14:35

The Demonix _ Hermit

3,478

answered Oct 11 '19 at 07:55

ankit satpati

1

For every natural number $x$, $\frac{1}{x} \le 1$

2 Answers2