Answer
See below.
Work Step by Step
1. Let $P(n)$ be the statement to be proved.
2. For $n=2$, we have $LHS=1$ hand shake (LHS represents the handshakes from counting, RHS from the formula) and $RHS=\frac{2(2-1)}{2}=1$, thus $LHS=RHS$ and $P(2)$ is true.
3. Assume $P(k), k\gt2$, is true, that is there are $\frac{k(k-1)}{2}$ hand shakes.
4. For $n=k+1$, we know that with the addition of one people, the increase of handshakes will be $k$, thus $LHS=k+\frac{k(k-1)}{2}=\frac{2k+k(k-1)}{2}=\frac{2k+k^2-k}{2}=\frac{k^2+k}{2}=\frac{k(k+1)}{2}=RHS$
5. Thus $P(k+1)$ is also true and we have proved the statement by mathematical induction.