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Step 1: Show that the statement holds for the basis case $n = 1$

$E(n) = n - 1$
$E(1) = 1 - 1 = 0$. A tree with one node has zero edges

Step 2: Assume that that summation is true up to n.

Step 3: Show that on the assumption that the summation is true for n, it follows that it is true for n + 1.

$E\left(n + 1\right) = n + 1 - 1$
$\Leftrightarrow E(n) + 1 = n$ When adding one node to a tree one edge is added as well
$\Leftrightarrow n -1 + 1 = n$
$\Leftrightarrow n = n$

QED