What does it mean "sequence with infinite range"

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I'm trying to understand this phrase

Find a sequence with infinite range that converges only to $0$.

What does it mean "sequence with infinite range"?

Thanks

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1 Answer

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Recall that a sequence is really a function from $\Bbb N$ into some set, $\Bbb R$ in this case (guessing by the choice of tags). If the sequence is $x\colon\Bbb N\to\Bbb R$ then we write, for convenience, $x(n)$ as $x_n$.

But since $x$ is a function, it has a range $\{x_n\mid n\in\Bbb N\}$ (where we forget about the enumeration). The sequence $x_n=0$, for example, is such that its range is a singleton, since all the $x_n$'s are equal to each other. On the other hand, the sequence $x_n=n$ has an infinite range, since $x_n=x_k$ if and only if $n=k$.

Your question asks you to find a sequence which converges to $0$ but does not have a finite range. Namely, infinitely many $x_n$'s are distinct from one another.

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