Let \(\xi _n\) and \(\eta _m\) be two sequences of random variables. Prove or give counterexamples to the following statements:
(a) If \(\xi _n\overset {d}{\to }\xi \) and \(\eta _m\overset {d}{\to }\eta \), then \(\xi _n+\eta _m\overset {d}{\to }\xi +\eta \).
(b) \(\xi _n\overset {P}{\to }\xi \) and \(\eta _m\overset {P}{\to }\eta \), then \(\xi _n+\eta _m\overset {P}{\to }\xi +\eta \).
(c) \((\xi _n,\eta _n)\overset {P}{\to }(\xi ,\eta )\) then \(\xi _n+\eta _n\overset {P}{\to }\xi +\eta \).
(\(\overset {d}{\to }\) and \(\overset {P}{\to }\) denote convergence in distribution and convergence in probability.)