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

Mathematics Asked by gaufler on December 4, 2020

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 $lim_{nrightarrow infty} Vert x_n -y_nVert$ exists

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

One Answer

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 $n,m>N$, $|(x_n-y_n)-(x_m-y_m)|=|(x_n-x_m)+(y_m-y_n)|leq |x_n-x_m|+|y_n-y_m|<c$.

Correct answer by Tsemo Aristide on December 4, 2020

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