Proof that doctors could relate to I am supposed to present a mathematical proof to a lecture hall full of doctors in order to show them how mathematicians think. I'm having trouble picking a proof that will be easily followed by people who haven't seen anything beyond basic calculus in years. Does anyone have any suggestions? I'm trying to think of something visual to keep their attention. 

Claim: $1+1/2+1/3+1/4+\ldots=\infty$. [sic!]
Proof:
\begin{align*}
&\,1+\frac{1}{2}+\underbrace{\frac{1}{3}+\frac{1}{4}}+\underbrace{\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}}+\underbrace{\frac{1}{9}+\ldots\frac{1}{15}+\frac{1}{16}}+\underbrace{\frac{1}{17}+\ldots+\frac{1}{31}+\frac{1}{32}}+\ldots\\
\geq&\,1+\frac{1}{2}+\underbrace{\frac{1}{4}+\frac{1}{4}}_{\text{2 terms}}+\underbrace{\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}}_{\text{4 terms}}+\underbrace{\frac{1}{16}+\ldots\frac{1}{16}+\frac{1}{16}}_{\text{8 terms}}+\underbrace{\frac{1}{32}+\ldots+\frac{1}{32}+\frac{1}{32}}_{\text{16 terms}}+\ldots\\
=&\,1+\frac{1}{2}+\left(2\times\frac{1}{4}\right)+\left(4\times\frac{1}{8}\right)+\left(8\times\frac{1}{16}\right)+\left(16\times\frac{1}{32}\right)+\ldots\\
=&\,1+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\ldots\\
=&\,\infty.
\end{align*}
Bam! Lecture hall full of doctors shocked! $\quad\blacksquare$

Then show numerical examples to further convince them on the intuitive level that the harmonic series actually does diverge, albeit very slowly. For example:
\begin{align*}
\sum_{k=1}^{10}\frac{1}{k}\approx&\,2\mathord.93,\\
\sum_{k=1}^{100}\frac{1}{k}\approx&\,5\mathord.19,\\
\sum_{k=1}^{1000}\frac{1}{k}\approx&\,7\mathord.49,\\
\vdots&\,\\
\sum_{k=1}^{10^6}\frac{1}{k}\approx&\,14\mathord.39,\\
\vdots&\,\\
\sum_{k=1}^{10^{100}}\frac{1}{k}\approx&\,230\mathord.84\\
\vdots&\,\\
\sum_{k=1}^{10^{(10^6)}}\frac{1}{k}\approx&\,2\mathord,302\mathord,586\\
\vdots&\,
\end{align*}
