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I am interested in the difference between constant (let's call it $c$) and random variable which is always a constant: $P(X = c) = 1$.

Is there any trap in thinking that it is the same? What about for example with the remark saying that: $P(X_n \neq 0) = 1, X_n \xrightarrow{\text{P}} X \implies \frac{1}{X_n} \xrightarrow{\text{P}} \frac{1}{X}$ Is this true if we take $X$ always equal to 1?

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No there is no "trap". If $\mathbb P(X = c) = 1$ then you can write that $X = 1$ almost surely. Your example work with $X_n = 1$, for all $n \in \mathbb N$.

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A constant random variable is a constant function $X:\Omega\to\mathbb R$. So the function is prescribed by $\omega\mapsto c$ for some $c\in\mathbb R$.

A random variable that is constant almost surely is a measurable function $X:\Omega\to\mathbb R$ with the property that $P(X=c)=1$.

If $X_n\stackrel{P}{\to} c$ where $c$ is a constant then also $X_n\stackrel{d}{\to} c$ and from this it follows that $f(X_n)\stackrel{d}{\to} f(c)$ for every continuous function $f$.

This on its turn implies that $f(X_n)\stackrel{P}{\to} f(c)$.

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