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I am not sure how to do the following question:

Let $f$ be a real-valued function defined for every $x$ in the interval $(0,1)$. Suppose there is a positive number $M$ having the following property: for every choice of a finite number of points $x_1, x_2, ... , x_n$ in the interval $(0,1)$, the sum $| f(x_1) + ... + f(x_n) | \leq M$ holds.

Let $S$ be the set of those $x \in (0, 1)$ for which $f(x) \neq 0$. Prove that $S$ is countable.

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    Hint: an uncountable set of real numbers has a non zero limitpoint. – 2017-02-28

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Hint: $$\{x\mid f(x)\ne 0\} = \bigcup_{n\in\mathbb N_+} \{x\mid |f(x)|>\tfrac1n\}$$ and each of the sets on the right-hand side must be finite.

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    I don't get why is each set on the right finite? – 2017-02-28
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    @JunaidAftab: If one of them is infinite, then it must contain either infinitely many _positive_ values or infinitely many _negative_ values. Suppose there are infinitely many positive values. Then adding up $nM$ of them would give a sum larger than $M$, contradicting your assumption. (The case for infinitely many negative values is exactly similar). – 2017-02-28