# $Vert x_{n} - y_{n} Vert$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

For every $$c>0$$, there exists $$N_1$$ such that $$n,m>N_1$$ implies that $$|x_n-x_m|, there exists $$N_2$$ such that $$n,m>N_2$$ implies that $$|y_n-y_m|, 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|.

Correct answer by Tsemo Aristide on December 4, 2020

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