# Let $(X, ||\cdot||)$ be a normed space. Let $\{x_n\}$ and $\{y_n\}$ be two Cauchy sequences in X.  Show that the seqience Show that the sequence $λ_n = ||x_n − y_n||$ converges.

Let $(X, ||\cdot||)$ be a normed space. Let $\{x_n\}$ and $\{y_n\}$ be two Cauchy sequences in X. Show that the seqience Show that the sequence $λ_n = ||x_n − y_n||$ converges.

Let $\epsilon>0$, and choose $N_0$ large enough so that for all $n,m\geq N_0$
$||x_n-x_m||< \epsilon, ||y_n-y_m||<\epsilon/2.$
By the reverse triangle inequality we have
$|\lambda_n-\lambda_m|=| ||x_n-y_n|| - ||x_m-y_m||| \leq ||x_n-y_n - (x_m-y_m)||$
$=||x_n-x_m - (y_n-y_m)|| \leq ||x_n-x_m||+ || y_n-y_m||$
$<\epsilon/2+\epsilon/2=\epsilon.$
Hence for all $n,m \geq N_0$ we have
$|\lambda_n-\lambda_m| < \epsilon,$
i.e. the sequene $\{\lambda_n\}$ is Cauchy, and thus it is a convergent series.

Erdos
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