# Mean Value Property of Harmonic Functions Proof Step

I'll only include the step that throws me off unless more info is requested, but this is from LC Evans PDEs book:

$$\displaystyle \lim_{t \to \, 0^+ } \left[ \frac{1}{n\,\alpha(n) \, t^{n-1}} \int_{ \partial{B(x,t)}} u(y) \, dS \; \right] = u(x)$$

where $\partial{B(x,t)}$ denotes a sphere centered at $x$ with radius $t$ and $n \, \alpha(n) \, t^{n-1}$ denotes the surface area of an n-sphere.

I thought maybe I could apply L'Hopital's, but it didn't seem to get me anywhere.

Thanks for any help.

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Choose $\epsilon > 0$ and $\delta >0$ so that $d(y,x) < \delta \implies |u(x)-u(y)|<\epsilon$. You are averaging things in $[u(x)-\epsilon, u(x) + \epsilon]$ so the average dwells in this interval as well. It's not a difficult estimate to perform.
@MattHancock u is continuous, so $u(y)$ is roughly equal to $u(x)$ when y is close to x. – Tommi Brander Apr 19 '13 at 12:08
@MattHancock Not to be obnoxious, Matt, but $\epsilon-\delta$ proofs are among the first thing one learns in analysis, and Evans' book will (should) be incomprehensible to you unless you take the time to learn basic analysis. – snarski May 21 '13 at 19:21