Is there such a proof that states that the Runge Phenomena will always occur when interpolating with higher order polynomials or is this just observed empirically?
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$\begingroup$The Runge Phenomenon does not always occur. You can interpolate, say, $f(x)= e^x$ using equally spaced nodes on any interval $[a,b]$, and the interpolating polynomials will converge to $f$ uniformly. (The same holds for any function whose Taylor series has infinite radius of convergence.)
When the Runge phenomenon does occur, it is a lot easier to observe empirically than to write down a rigorous proof that it happens. This is why the textbooks on numerical analysis tend to do the former and not the latter.
For the classical example of intepolating $f(x)=1/(1+x^2)$ on $[-5,5]$ by equally spaced nodes, David Speyer posted a complete proof of divergence at $x=4$.
$\endgroup$ $\begingroup$Runge Phenomena doesn't occur for all functions. For a detailed analysis on polynomial interpolation at equidistant points you can have a look here
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