in the below picture you can see my problem. Honestly, I am in a probability course and this isn't even remotely similar to anything we've covered...I'm not totally sure where to even begin. If anyone could help me decipher what this means I would really appreciate it.
Since we are asked if $\lim_{n \to \infty} P(M_{n})=0$ is always true,
if we can show one counter example, than that will suffice to answer the question.
Suppose {${X_{n}}$} is a sequence of random variables such that $(P(X_{n})=1)=1$
Then, $\lim_{n \to \infty} P(M_{n})=1$
Hence, the statement is not always true.
If you consider the definition of "support" to mean that any value in the support MUST have a non zero probability of occurring, then lets take a closer look. If n=0 then it is true ( as only zero can occur so mode is zero).
But as $n \to \infty$, the probability of being zero must decrease to make room for the probability of having other outcomes. Hence the probability that we have zero approaches zero, and hence the probability that the mode is zero also approaches zero, so it is false.