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I am trying to do the following question, because I think it is very interesting as it formalises the idea that the tangent space is practically indistinguishable from the surface if one looks in a very small neighbourhood of a point. The question is from "differential Geometry of Curves and Surfaces" by Tapp.

My first idea was to show that $limlimits_ntoinftyfracp_n-p=:vin T_pS$ but I don't see why/if this limit exists.

I would be very greatful for some hints or even a solution.
Thanks a lot in advance!:)

Definition of regular surface:

I tried to use $p_n-p=sigma(x_n)-sigma(x))=d_psigma(x_n-x)+o(|x_n-x|)$ to rewrite $langle N,fracp_n-prangle$

It may very well be that $lim_n to infty fracp_n-p$ does not exist. (To see this, consider an oscillation near a point.)

But by taking subsequences, we can prove the result, even if the limit above does not exist by itself. Fix a subsequence $p_n_i$ of $p_n$.

First, taking a parametrization $varphi$ near $p$, we have that $p_n_i$ is in that chart for $i$ sufficiently big. Letting $q_n_i=varphi^-1(p_n_i)$, we know that
beginalign
fracp_n_i-p&=fracvarphi(q_n_i)-varphi(q)-dvarphi(q_n_i-q)+dvarphi(q_n_i-q)q_n_i-qcdot fracq_n_i-q \
&=fracvarphi(q_n_i)-varphi(q)-dvarphi(q_n_i-q)q_n_i-q +dvarphileft(fracq_n_i-qq_n_i-qright)cdot fracq_n_i-q.
endalign
By passing to subsequences yet again (twice), this converges to
$Kcdot dvarphi(h)$ for some $h in mathbbR^2$, $K in mathbbR$. Therefore, the limit of this subsequence $fracp_n_i_j-p$ of $p_n_i$ belongs to $T_pS$ and it follows that
$$lim_j to infty leftlangle N, fracp_n_i_j-prightrangle=0.$$
Letting $a_n=leftlangle N, fracp_n-prightrangle$, we have thus proved that every subsequence of $a_n$ has a further subsequence converging to $0$. This proves that $a_n$ converges to zero. (This is a general fact about sequences - see here, for example.)