How can a sequence of random variables converges in probability to a constant? [duplicate]

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I have a difficult time understanding how can a sequence of random variables, lets say {$z_i$}, converge in probability to a constant.

Is that the mean of the realization of all random variables in the sequence converges to a number? (If yes, isn't it just the central limit theorem?)

Thank you.

asked Sep 14, 2016 at 1:30

Leopold W.

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$\begingroup$ $z_n=0$ with probability $1/n$, z=1 else. What happens to z as n goes to infinity? $\endgroup$
– VCG
Commented Sep 14, 2016 at 1:47
$\begingroup$ Thanks for the comment, @VCG. I understand the mechanics of convergence in probability. But I can't follow the intuition. What does $n$ represent exactly? Sample size or just an index? If it is an index, what's the difference between n=5 and n=100000? $\endgroup$
– Leopold W.
Commented Sep 14, 2016 at 1:54
$\begingroup$ There are a few interpretations statlect.com/asymptotic-theory/sequences-of-random-variables $\endgroup$
– VCG
Commented Sep 14, 2016 at 2:07
$\begingroup$ Some duplicates: stats.stackexchange.com/questions/134701/…, stats.stackexchange.com/questions/337788/…, $\endgroup$
– kjetil b halvorsen ♦
Commented Aug 16, 2020 at 17:44

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