sequences convergence

Question

Let $\{x_n\}$ be a sequence of real numbers. Prove that $x_n \to x$ if and only if $$x = x_1 + \sum_{k=1}^{\infty}(x_{k+1} - x_k)$$ I have initially tried solving by expanding the bottom part all the terms in the sigma expansion gets cancelled and we are left out with a single term, How to show the convergence criteria.Should I prove its cauchy sequence or any other approach?

Answer 1

If $x_n\rightarrow x$. Let $y_n=x_1+\sum_1^n (x_{k+1}-x_k)=x_{n+1}\rightarrow x$. So the underlying infinite series converges to $x$ and conversely...