I'm grateful to any help.

I'm supposed to show that $\displaystyle x_{n}-x_{n-1} \rightarrow 0$ as $\displaystyle n \rightarrow \infty$ if $\displaystyle \{x_n\}$ converges.

Suppose the sequence converges to x. Since $\displaystyle \{x_n\}$ converges, $\displaystyle \{x_{n-1}\}$ also converges.

This implies that for every $\displaystyle \epsilon \textgreater 0$, there exist integers $\displaystyle N_1, N_2$ s.t $\displaystyle N_1 \textgreater n$ and $\displaystyle N_2 \textgreater n-1$ implying absolute value of $\displaystyle (x_n-x) \textless \epsilon$ and absolute value of $\displaystyle x_{n-1}-x \textless \epsilon$.

Am I on the right track to solve this problem? Here, do I need to use the definition of limit to show that $\displaystyle lim (b_n -b_{n-1})=0$ as $\displaystyle n \rightarrow \infty$