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I think the limit is $1$, because $\left\lceil \frac{1}{n}\right\rceil=1$ for all $n$, but it seems counter intuitive for some reason. If we crudely "replace $n$ with $\infty$" we get $\lceil 0\rceil=0$ (also, if it was $0$ I can't see a way of proving it).

Is this due to the discontinuous character of $\lceil x\rceil?$

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You’re right, and it shows the undependibility of intuition; or alternatively, that the discontinuous functions are the ones that go against intuition. – Lubin Mar 4 '13 at 15:27
up vote 12 down vote accepted

You are exactly correct - the limit is $1$, because the sequence is constant. The limit of $\left\lceil-\frac{1}{n}\right\rceil$ as $n\to\infty$ is $0$, and this demonstrates the discontinuity of $\lceil x\rceil$.

If the limit were $0$, the sequence would need to become arbitrarily small as $n$ increased. So for example, there should be some $n$ such that $\left\lceil\frac{1}{n}\right\rceil<\frac{1}{2}$, but there isn't, because $\left\lceil\frac{1}{n}\right\rceil=1$ for all $n$.

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