What does ∈ mean in the exponent? I'm having troubles understanding the following proof:
$$
\begin{align*}
&amp;\text{Proof: } \forall \epsilon \in \mathbb{R}^+, \forall a \in \mathbb{Z}^+, n^\epsilon \gg \log_a(n) \\
&amp;\Longrightarrow \ln(a(n)) = \log_2(e)\ln^2(n) \ll n^{0.01} = \ln(d(n)) \\
&amp;\Longrightarrow a(n) \ll d(n)
\end{align*}
$$
The topic is Big O comparisons. The proof is about $a(n)= n^{\log_2 n}$ and $d(n) = e^{n^{0.01}}$. The proof is very heavy in mathematical notation. I'm not sure what ∈ means in the example.  I know that it denotes set membership in a more usual case, but have never seen it in the exponential position.
That's not $\in$ but $\epsilon$: that is, a lowercase Greek letter epsilon. It's normally written $\varepsilon$ instead for clarity, but this author chose to use the "lunate" form for unknown reasons.

In this case, epsilon is being used as a shorthand for "an arbitrarily small but positive real number". This should probably be explained in the proof, but it's a fairly common shorthand in mathematics (for example, delta-epsilon proofs, floating-point epsilon…).

In other words, it's saying that the log of $n$ is always asymptotically smaller than $n$ raised to any positive power. In the next line it substitutes in a definite value, 0.01, for epsilon.