Does $ \left\lceil \frac{1}{n}\right\rceil\to 0$ or $1$ as $n\to\infty ?$

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 $\lceil x\rceil?$

Solution 1:

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 $n$.