jmc

We find

$$\frac{1}{n}-\frac{1}{n+1}=\frac{1}{n(n+1)}$$

So we want $\frac{1}{n(n+1)}<\frac{1}{10}$, or $n(n+1)>10$. We see that $2(3)=6<10$, while $3(4)=12>10$. So the least possible value is $\boxed{3}$.