Show that $\int_{0}^{1}{\frac{\sin{x}}{x}\mathrm dx}$ converges As title says, I need to show that the following integral converges, and I can honestly say I don't really have an idea of where to start. I tried evaluating it using integration by parts, but that only left me with an $I = I$ situation. 
$$\int \limits_{0}^{1}{\frac{\sin{x}}{x} \mathrm dx}$$

Take the function 
$
f(x)=
\left \{
\begin{array}{cc}
\frac{\sin x}{x} & x \in (0,1]\\
1& x=0
\end{array}
\right .
$
We observe that this function is continuous on $[0,1]$ hence Riemann integrable 
on $[0,1]$
