- #1
Shoelace Thm.
- 60
- 0
Homework Statement
Prove that [itex] (\mathbb{R},d) [/itex], [itex] d = \frac{\mid x - y \mid}{1 + \mid x - y \mid} [/itex] is a complete metric space.
Homework Equations
The Attempt at a Solution
If [itex] d_u = \mid x - y \mid [/itex], then I can prove this for the Cauchy sequences in [itex] (\mathbb{R},d) [/itex] that are also Cauchy in [itex] (\mathbb{R},d_u) [/itex]. But there may be Cauchy sequences in [itex] (\mathbb{R},d) [/itex] that are not Cauchy in [itex] (\mathbb{R},d_u) [/itex].