I do not want to argue against

**Opalg** because I am a noob compared to him in analysis

, but I offer a suggestion.

I think this problem is better solved using the concepts of limsup's and liminf's.

Let

be the length of the interval. Let

.

Opalg hath shown us that

.

Then he says

by

.

But we need to know the limit exists.

So a way around this problem is to perhaps say first take

and then

.

But the problem is

depends on

and so how we know that

as

?

Maybe I am missing something but I think that if we use limsup's and liminf's the solution is much smoother. Because

and so

. However,

. Thus, we have shown

but

. This means that

is a convergent sequence with

.

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