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## Homework Statement

[itex]\mathbf{D1:}\forall\varepsilon>0,\exists K\epsilon\mathbb{N},\forall n\epsilon\mathbb{N},n\geq K\Longrightarrow|x_{n}-x|<\varepsilon[/itex]

[itex]\mathbf{D2:}\forall\rho>0,\exists M\epsilon\mathbb{N},\forall n\epsilon\mathbb{N},n>M\Longrightarrow|x_{n}-x|\leq\rho[/itex]

Show these two definitions of convergence (of a sequence xn) are equivalent by showing both D1 implies D2, and D2 implies D1.

## Homework Equations

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## The Attempt at a Solution

To show D1 ==> D2, i just took M = K - 1, where K was from D1. But it didn't seem to get me anywhere since i can show

[itex]n>M\Longrightarrow|x_{n}-x|< \rho[/itex],

but this does not encompass the possibility of ≤ ρ. So basically i have no idea how to do this.

Similarly, i need something to get me started on D2 ==> D1. How do I go about this?

I'm sure i'm overthinking it as I've done much more complicated proofs before, and I've done all other questions on this assignment without help. I just honestly don't know how to do this one. Thanks in advance!