State and prove Markov's inequality and Chebyshev's inequality, and deduce the weak law of large numbers.

If $X$ is a random variable with mean zero and finite variance $\sigma^{2}$, prove that for any $a>0$,

$$P(X \geqslant a) \leqslant \frac{\sigma^{2}}{\sigma^{2}+a^{2}}$$

[Hint: Show first that $P(X \geqslant a) \leqslant P\left((X+b)^{2} \geqslant(a+b)^{2}\right)$ for every $b>0$.]