open interval and differentible and sequences.. proof help

[lawilson8510]

If I is an open interval, f is differentiable on I and a is in I, then there is a sequence a_n in I \{a} such that a_n->a and f '(a_n)->f '(a).

I need a proof

 

[pka]

If I is an open interval, f is differentiable on I and a is in I, then there is a sequence a_n in I \{a} such that a_n->a and f'(a_n)->f'(a).
I need a proof

Don’t you mean that you need to do a proof?
This requires a very careful picking a sequence of points by repeated use of the mean value theorem.
There is a \(\displaystyle a_1\in (a,c_1)\) such that \(\displaystyle f{^'}(a_1)=\frac{f(c_1)-f(a)}{c_1-a}\).
You note that fraction is a difference quotient approximating \(\displaystyle f{^'}(a)\).
You must take care that \(\displaystyle (a_n)\) converges to \(\displaystyle a\).