# let $c$ be a positive number. Prove that the set $S$ = {$c, 2c, 3c, ..., nc, ...$} is not bounded above

My answer is not quite robust enough, but this is what I'm thinking... Think I can state it clearly for any number $$c \geq 1$$ but I know that doesn't cover any number $$0\lt c \le 1$$

let $$c$$ be a positive number. Prove that the set $$S$$ = {$$c, 2c, 3c, ..., nc, ...$$} is not bounded above

Let's assume that $$c$$ is equal to $$1$$. In the set $$S$$, we can see that the result would be $$S$$ = {$$1, 2, 3,..., n,...$$} which has no upper bound.

Since any positive number $$c > 1$$ would behave the same way, we can see there would be no upper bound in this set for any number $$c$$ $$\geq 1$$

Naturally this would also hold true for any number greater than $$0$$

Maybe another way is to take a look at

$$f \colon \mathbb{N} \rightarrow S$$ with $$f(n) = cn$$. Then obviously $$\mathrm{Im} f = S$$, and you only need to show that $$f$$ is unbounded.

So assuming there is an $$M \in \mathbb{R}$$ s.t. $$f(n) \leq M \quad \forall \, n \in \mathbb{N}$$.

Then $$cn \leq M$$ for all $$n \in \mathbb{N}$$ which is equivalent to $$c \leq \frac{M}{n}$$ for all natural numbers.

Choosing now $$n \geq \frac{2M}{c}$$ we observe that $$c \leq \frac{c}{2}$$ must hold, which is a contradiction.

Let $$M$$ be any positive real number, and note that $$M \leq c\lceil{M/c}\rceil$$, where $$\lceil{-}\rceil$$ is the ceiling function, so $$c\lceil{M/c}\rceil \in S$$. That is to say, for any positive real number $$M$$ we can find a number $$c\lceil{M/c}\rceil$$ in $$S$$ greater than $$M$$, so $$S$$ is not bounded above.

Let $$k$$ denote the least upper bound so some $$n\in\Bbb N$$ satisfies $$nc>k-c$$, whence $$(n+1)c=nc+c\ge k$$, a contradiction.

• I'm having some trouble following this one (my apologies). Would something like this work: $\frac{nc}{(n+1)c} \gt 1$? Oct 24, 2019 at 2:54
• @B.Hoffman I don't see how you derived it. The point is if $k$ is minimal any real $<k$, including $k-c$, is exceeded by some element of $S$.
– J.G.
Oct 24, 2019 at 5:29