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Are the following statements true?

Suppose we have a sequence of constants $c_n \rightarrow c$. Then:

  1. If $X_n \stackrel{a.s.}{\rightarrow} X$, then $c_nX_n \stackrel{a.s.}{\rightarrow} cX$.
  2. If $X_n \stackrel{p}{\rightarrow} X$, then $c_nX_n \stackrel{p}{\rightarrow} cX$.
  3. If $X_n \stackrel{d}{\rightarrow} X$, then $c_nX_n \stackrel{d}{\rightarrow} cX$.

For example, I'd like to say that $\frac{n}{n-1} X_n \stackrel{a.s.}{\rightarrow} X$, when $X_n \stackrel{a.s.}{\rightarrow} X$. The closest thing I could find was Slutsky's lemma which holds when $c_n$ is a sequence of random variables (but not constants) such that $c_n \stackrel{d}{\rightarrow} c$.

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    A constant can be seen as a random variable, for what it's worth. – 2017-02-01
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    do you want proofs or references or what – 2017-02-01
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    Clement's answer is fine - thanks, did not know that – 2017-02-01

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