Let be an infinite sequence of reals. Show that if is convergent then we must have . If , must be convergent?
What have you tried? Note that $\displaystyle \left|a_n-a_{n-1}\right|\le\left|a_n-L\right|+\left|a_{n-1}-L\right|$. For your second question, what about $\displaystyle a_n=n+\frac{1}{n}\implies a_{n}-a_{n-1}=\frac{1}{n}-\frac{1}{n-1}$?