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I want to maximize

$${T(\epsilon)=\max\left\{\log_{(1/y)}\left(|\sqrt{2}-x/y|\right):x,y\in\mathbb{N}, |\sqrt{2}-x/y|<\epsilon\right\}}$$

Edit: I changed the inequality, it was supposed to be greater than

Edit 2: Perhaps it should be less than

When I used my code I tried (for $\epsilon=.0001$)

NMaximize[{x, y} \[Element] Integers && Log[1/y, Sqrt[2] - x/y] && 
  RealAbs[Sqrt[2] - x/y] >= .0001, {x, y}]

But instead I get:

NMaximize::nnum: The function value -False is not a number at {x,y} = {0.918621,0.716689}.

NMaximize::nnum: The function value -False is not a number at {x,y} = {0.918621,0.716689}.

NMaximize::nnum: The function value -False is not a number at {x,y} = {0.918621,0.716689}.

How do we fix this and create $T(\epsilon)$ in my code.

I want to maximize

$${T(\epsilon)=\max\left\{\log_{(1/y)}\left(|\sqrt{2}-x/y|\right):x,y\in\mathbb{N}, |\sqrt{2}-x/y|<\epsilon\right\}}$$

Edit: I changed the inequality, it was supposed to be greater than

Edit 2: Perhaps it should be less than

When I used my code I tried (for $\epsilon=.0001$)

NMaximize[{x, y} \[Element] Integers && Log[1/y, Sqrt[2] - x/y] && 
  RealAbs[Sqrt[2] - x/y]<= .0001, {x, y}]

But instead I get:

NMaximize::nnum: The function value -False is not a number at {x,y} = {0.918621,0.716689}.

NMaximize::nnum: The function value -False is not a number at {x,y} = {0.918621,0.716689}.

NMaximize::nnum: The function value -False is not a number at {x,y} = {0.918621,0.716689}.

How do we fix this and create $T(\epsilon)$ in my code.

I want to maximize

$${T(\epsilon)=\max\left\{\log_{(1/y)}\left(|\sqrt{2}-x/y|\right):x,y\in\mathbb{N}, |\sqrt{2}-x/y|>\epsilon\right\}}$$

Edit: I changed the inequality, it was supposed to be greater than

When I used my code I tried (for $\epsilon=.0001$)

NMaximize[{x, y} \[Element] Integers && Log[1/y, Sqrt[2] - x/y] && 
  RealAbs[Sqrt[2] - x/y] >= .0001, {x, y}]

But instead I get:

NMaximize::nnum: The function value -False is not a number at {x,y} = {0.918621,0.716689}.

NMaximize::nnum: The function value -False is not a number at {x,y} = {0.918621,0.716689}.

NMaximize::nnum: The function value -False is not a number at {x,y} = {0.918621,0.716689}.

How do we fix this and create $T(\epsilon)$ in my code.

I want to maximize

$${T(\epsilon)=\max\left\{\log_{(1/y)}\left(|\sqrt{2}-x/y|\right):x,y\in\mathbb{N}, |\sqrt{2}-x/y|<\epsilon\right\}}$$

Edit: I changed the inequality, it was supposed to be greater than

Edit 2: Perhaps it should be less than

When I used my code I tried (for $\epsilon=.0001$)

NMaximize[{x, y} \[Element] Integers && Log[1/y, Sqrt[2] - x/y] && 
  RealAbs[Sqrt[2] - x/y] >= .0001, {x, y}]

But instead I get:

NMaximize::nnum: The function value -False is not a number at {x,y} = {0.918621,0.716689}.

NMaximize::nnum: The function value -False is not a number at {x,y} = {0.918621,0.716689}.

NMaximize::nnum: The function value -False is not a number at {x,y} = {0.918621,0.716689}.

How do we fix this and create $T(\epsilon)$ in my code.