MEMO 2019 T4

Prove that every integer from $1$ to $2019$ can be represented as an arithmetic expression consisting of up to $17$ symbols $2$ and an arbitrary number of additions, subtractions, multiplications, divisions and brackets. The $2$ 's may not be used for any other operation, for example, to form multidigit numbers (such as $222$ ) or powers (such as $2^2$ ).

Valid examples: $$\left((2\times 2+2)\times 2-\frac{2}{2}\right)\times 2=22 \;\;, \; (2\times2\times 2-2)\times \left(2\times 2 +\frac{2+2+2}{2}\right)=42$$ Proposed by Stephan Wagner, Austria