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.
Answer
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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