Does this particular limit exist? $f(x)$ is a continuous, decreasing function from the positive reals to the positive reals.  
Let $$a_N = \sum_{1}^Nf(n) - \int_{1}^N f(x)dx.$$
Does the limit of $a_N$ always exist? Prove it or give a counterexample.
I would like some hints only.  
Thanks.

Write $a_N$ in the form
$$a_N=f(N)+\sum_{k=1}^{N-1}\left(f(k)-\int_k^{k+1} f(t)\>dt\right)\ .$$ Here the $\lim_{N\to\infty} f(N)$ exists since $f$ is monotonically decreasing with $0$ as lower bound. On the other hand $$c_k:=f(k)-\int_k^{k+1} f(t)\>dt$$ can be estimated by
$$0\leq c_k\leq f(k)-f(k+1)\ .$$
It follows that the partial sums $s_N:=\sum_{k=1}^{N-1} c_k$ are monotonically increasing and that
$$s_N\leq\sum_{k=1}^{N-1}\bigl(f(k)-f(k+1)\bigr)=f(1)-f(N)\leq f(1)\qquad(N\geq1)\ .$$
Therefore the $\lim_{N\to\infty} s_N$ exists as well, and so does the limit of the $a_N=f(N)+s_N$.
