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Step 1: prove for $n = 1$
1 < 2
$n+1 < 2 \cdot 2^n$
$n < 2 \cdot 2^n - 1$
$n < 2^n + 2^n - 1$
The function $2^n + 2^n - 1$ is surely higher than $2^n - 1$ so if
$n < 2^n$ is true (induction step), $n < 2^n + 2^n - 1$ has to be true as well.
Is this valid argumentation?