Question

Suppose that (x_n) and (y_n) are Cauchy sequences in a metric space (X,d). Show that the numerical sequence a_n=d(x_n,y_n) converges.

Answer 1

$d(x_{n} , x_{m}) + d(y_{n}, y_{m})$$d(x_{n} , x_{m}) - d(y_{n}, y_{m})$According to proof, the sequence ($r_{n}$)=(d($x_{n}$,$y_{n}$)) is Cauchy. See the proof below.

What does it mean for an expression to be Cauchy?

A Cauchy statement, in the language of mathematics, is a progression of components that are arbitrarily near to one another.

What is the proof for the above Cauchy expression?

Let ε > 0

N ∈ N in a way that

Whenever n, m > N, the result is:

|$r_{n} - r_{m}$|

= |$d(x_{n} , y_{n}) - d(x_{m}, y_{m})$ |

= | ($d(x_{n} , y_{n}) - d(x_{m}, y_{n}) + d(x_{m}, y_{n}) - d(x_{m}, y_{m})$|

≤ |$d(x_{n} , y_{n}) - d(x_{m}, y_{n}) | + | d(x_{m} , y_{n}) - d(x_{m}, y_{m})$ |

≤ $d(x_{n} , x_{m}) - d(y_{n}, y_{m})$ < ε

QED
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