Convergence of a sequence of Random variables I have recently been studying the convergence of a sequence of random variables. However, 
Let $\left\{ X_{n}\right\} _{1}^{\infty}$ be a sequence of random
variables defined on $\left(\Omega,F,P\right)$ where the range of
each term $X_{n}$ is the singleton set $\left\{ 1+\frac{1}{n}\right\} .$
First, I wish to be able to find  $\left\{ F_{X_{n}}\right\} _{1}^{\infty}$ and $\left\{ f_{X_{n}}\right\} _{1}^{\infty}$. Secondly, I'd like to find out whether or not   
a. $\left\{ X_{n}\right\} _{1}^{\infty}$ converge
in distribution  
b.  $\left\{ X_{n}\right\} _{1}^{\infty}$ converge
in probability  
c.  $\left\{ X_{n}\right\} _{1}^{\infty}$ converge
in almost sure sense  
d. $\left\{ X_{n}\right\} _{1}^{\infty}$ converge
in $p$-th moment for $p\geq1$.    
I know the definitions of convergence in distribution, probability and almost sure convergence. However, I do not know what convergence in $p$-th moment mean.
Please, any help on how to  begin will be appreciated.
Thanks.
 A: You have the sequence of constant variables $X_n = 1+1/n$ ($X_n (\omega) = 1+1/n$ for any $\omega \in \Omega$). One should expect that it converges to $1$ as $n \to \infty$, in all modes.
For a) you need to show that
$$
\mathop {\lim }\limits_{n \to \infty } P(1 + 1/n \le x) = P(1 \le x),
$$
for any $x \neq 1$.
For b) you need to show that
$$
\mathop {\lim }\limits_{n \to \infty } P(|(1 + 1/n) - 1| > \varepsilon ) = 0,
$$ 
for any $\varepsilon > 0$.
For c) you need to show that
$$
P(\lim _{n \to \infty } (1 + 1/n) = 1) = 1.
$$
For d) you need to show that
$$
\mathop {\lim }\limits_{n \to \infty } {\rm E}|(1 + 1/n) - 1|^p  = 0.
$$
EDIT: Concerning the sequence of distribution functions $(F_{X_n})$, note that
$$
F_{X_n } (x): = P(X_n  \le x) = P(1 + 1/n \le x).
$$
The distribution function of the limit $X=1$ is given by $F_X (x)=P(1 \leq x)$, $x \in \mathbb{R}$. It is important to remember that $X_n$  converges in distribution to $X$ if and only if $F_{X_n}(x) \to F_X (x)$ (as $n \to \infty$) for any $x \in \mathbb{R}$ which is a continuity point of $F_X$. (Note that, in our case, $x$ is a continuity point of $F_X$ if and only if $x \neq 1$.) 
Finally, $X_n$ (like any other discrete random variable) does not have a probability density function; hence, $f_{X_n}$ does not exist. 
