How to prove theorem 5.2.3 per below?

[5.2.2 given for reference]

Theorem 5.2.3. Suppose X" converges to X in. distribution and Y\" converges in probability to D. Then X" +Yn converges to X in distribution. The proof is similar to that of Theorem 5.2.2 and is left to Exercise 5.2.12; We often use this last result as follows. Suppose it is diicult to show that X\" converges to X in distribution, but it is easy to show that Y\" converges in distribution to X and that X\" Y" converges to 0 in probability. Hence, by this lest theorem, x\" = 1",, + (L, Y\") 3 X, as desired. r! v.1 :- Theorem 5.2.2. If Xn converges to the constant b in distribution, then Xn con- verges to b in probability. Proof: Let 0 be given. Then lim PIXn - b|