Show that if $\{x_{n}\}$ and $\{y_{n}\}$ are Cauchy sequences in X then $d(x_{n},y_{n})$ converges in $\mathbb{R}$.

Hint: use triangular inequality $$|d(x_n,y_n)-d(x_m,y_m)|\leq |d(x_n,y_n)-d(x_n,y_m)|+|d(x_n,y_m)-d(x_m,y_m)|,$$ then the reversed triangular inequality $|d(x,y)-d(z,y)|\leq d(x,z)$.


This is essentially Davide Giraudo's approach, but somewhat shorter: The triangle inequality gives $$d(x_m,y_m)\leq d(x_m,x_n)+d(x_n,y_n)+d(y_n,y_m)$$ or $$d(x_m,y_m)-d(x_n,y_n)\leq d(x_m,x_n)+d(y_n,y_m)\ .$$ As the right side is symmetric in $m$ and $n$ we have in fact $$\bigl|d(x_m,y_m)-d(x_n,y_n)\bigr|\leq d(x_m,x_n)+d(y_n,y_m)\ .$$

Tags:

Sequences And Series

Real Analysis

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