Switching limits: $n \rightarrow \infty$ for $n\rightarrow 0$ I feel like this question may have already been asked, but despite my searches, I could not find it.
I am looking to prove that $\lim\limits_{\epsilon \rightarrow 0} \int_{[b, b+\epsilon]} g=0$  for an integrable function $g$.  To do so, I would like to use the Dominated Convergence Theorem, as $g$ is my integrable dominating function.  However, I realize that the DCT is defined for the limit as $n$ goes to infinity.  I am looking to somehow switch this $\epsilon$ for $\frac{1}{k}$ and then take $k$ to infinity and use DCT.  However, I feel that it cannot be so simple.  How do I go about applying DCT to limits which are not going to infinity?
Any hints would be appreciated.  

Your approach of setting $\epsilon = 1/k$ can be made to work.  However, you need to make some adjustments to your argument:


*

*Note that it suffices to show that $\int_{[b,b+\epsilon]}|g|\to 0$

*Note that for $\epsilon \in [1/(k+1),1/k]$, we have
$$ 
\int_{[b,b+1/(k+1)]}|g| \leq
\int_{[b,b+\epsilon]}|g| \leq 
\int_{[b,b+1/k]}|g|
$$


Now, use your argument to show that $\int_{[b,b+1/k]}|g| \to 0$, and by the above arguments that's enough.
