$\Vert x_{n} - y_{n} \Vert $is a cauchy sequence in $\mathbb{F}$

Question

Given $X$ is a normed linear space over the field $\mathbb{F}$ and $(x_{n}), (y_{n})$ be Cauchy seuqences in $X$ then $\Vert x_n - y_n \Vert$ is a Cauchy sequence in $\mathbb{F}$ and consequently the $\lim_{n\rightarrow \infty} \Vert x_n -y_n\Vert$ exists

I know that this can be stated using the triangle inequality but I am not able to see how to use it here

Answer 1

For every $c>0$, there exists $N_1$ such that $n,m>N_1$ implies that $\|x_n-x_m\|<c/2$, there exists $N_2$ such that $n,m>N_2$ implies that $\|y_n-y_m\|<c/2$, take $N=\max(N_1,N_2)$ if $n,m>N$, $\|(x_n-y_n)-(x_m-y_m)\|=\|(x_n-x_m)+(y_m-y_n)\|\leq \|x_n-x_m\|+\|y_n-y_m\|<c$.