Riya Andrews

2022-09-06

Give an indirect proof for

$d({x}_{n},{y}_{n})\to d(x,y)$

$d({x}_{n},{y}_{n})\to d(x,y)$

Colin Dougherty

Beginner2022-09-07Added 8 answers

An "indirect proof" usually starts with "Suppose not." So to go about this, suppose $d({x}_{n},{y}_{n})$ does not tend to $d(x,y)$ as ${x}_{n}\to x$, ${y}_{n}\to y$. Then there is an ${\u03f5}_{0}>0$ such that there exists no $N$ such that for all $n\ge N$,

$|d({x}_{n},{y}_{n})-d(x,y)|\ge {\u03f5}_{0}$

Now use the triangle inequality as in the direct proof to get a contradiction.

$|d({x}_{n},{y}_{n})-d(x,y)|\ge {\u03f5}_{0}$

Now use the triangle inequality as in the direct proof to get a contradiction.

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