Give an indirect proof for d ( x n , y n ) → d ( x , y )

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 → x, y n → y. Then there is an ϵ 0 &gt; 0 such that there exists no N such that for all n ≥ N, | d ( x n , y n ) − d ( x , y ) | ≥ ϵ 0 Now use the triangle inequality as in the direct proof to get a contradiction.

"Give an indirect proof for d ( x..." was answered by our experts

Riya Andrews

Answered question

2022-09-06

Give an indirect proof for

d (x_{n}, y_{n}) \to d (x, y)

Answer & Explanation

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)

x_{n} \to x

y_{n} \to y

. Then there is an

ϵ_{0} > 0

such that there exists no

N

such that for all

n \geq N

| d (x_{n}, y_{n}) - d (x, y) | \geq ϵ_{0}

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

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