Pointwise limit of the indicator function: $\lim_{c \to \infty}1_{\{X<kc\}}$

Question

I want to find the pointwise limit of this indicator function:

$$\lim_{c \to \infty}1_{\{X<kc\}}$$ where $k>0$ is a parameter and $1$ is an indicator function and $X$ is finite random variable.

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One of my friends says it would be $\lim_{c \to \infty}1_{\{X<kc\}}=1_{\{x<\infty\}}=1$. Please provide me correct answer with explanation. Thank you.

Answer 1

Given $t$ in the domain of $X$, take $M$ so that $M>\frac{X(t)}{k}$. This is possible because $X$ is finite and $k$ is fixed. Then, for all $c\geq M$, we have $1_{X<kc}(t) = 1$ because $X(t)<Mk\leq ck$. Therefore $\lim_{c\to\infty}1_{X<kc}(t)=1$ for each $t$, which means the pointwise limit $\lim_{c\to\infty}1_{\{X<kc\}}$ is $1$.