Very tricky problem from Spivak's Calculus, Ch. 4 & 5: Graphs/Limits

[zenterix]

The main proof as you call it is potentially important, as it formalises the argument. But, without the side proof it means nothing.

That proposition must be false. The problem is that a small reduction in $x$ after a string of zeroes may cause a long string of nines. Using the double digit idea solves this problem.

The proposition should actually be with a $\leq$, ie $a-\delta \leq f(x)$. Is this what you mean?

 

[PeroK]

The proposition should actually be with a $\leq$, ie $a-\delta \leq f(x)$. Is this what you mean?

You're right. But, I don't see how that proposition helps.

 

[zenterix]

This is effectively what you need to prove. Otherwise your ϵ−δ is just window-dressing!

It's the proof of this, which was a step in the main proof. It's required for the latter.