how to show that a function is unbounded?

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How to prove that the function $f:(0,2)\to\mathbb{R}, f(x)=\frac{1}{x}$ is unbounded.

I know for a function is unbounded if: $\forall M>0 \exists x\text{ such that }|f(x)|>M$

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2 Answers

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If it was bounded by, let us say, $C$, then $$ \left| x \frac{1}{x} \right| \leq C |x|, $$ and hence $\lim_{x \to 0} 1 = 0$.

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$$\lim_{x\to 0^+}\frac{1}{x}=+\infty $$ therefore $$\forall M\geq 0, \exists \delta>0: |x|<\delta\implies |f(x)|> M$$

in particular $$\forall n\in\mathbb N, \exists x_n: |f(x_n)|>n$$ therefore $f$ is unbounded.

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