Euler characteristic by induction

Setup. Induct on the number of edges $E$.

Base case. A tree has $F=1$ (only outer face), $V-E+1=1$ (since $E=V-1$), so $V-E+F=V-(V-1)+1=2$. ✓

Inductive step. If the graph has a cycle, remove an edge from the cycle: the graph stays connected, $E$ decreases by $1$, and $F$ decreases by $1$ (two faces merge). Both sides of $V-E+F=2$ change by $-1+1=0$, preserving the identity.

Conclusion. $V-E+F=2$ for connected planar graphs.