I am trying to prove that there exists a $M$ such that for every $e$:
$$\sum_{i = n}^n P(|X_i - Y_i|> M) < e$$
By Markov inequality and by the fact that $E|X_i - Y_i| \leq c (= O(1))$ uniformly for every $i$, we have that
$$\sum_{i = n}^n P(|X_n - Y_n|> M) < cnM^{-1}$$.
Now:
- if $e = 1$, I would set M = 2nc,
- if $e = 10$ then, $M = \frac{1}{10}nc$, etc...
Generally, if somebody gives me a $e$ I would choose $M = \frac{1}{e}nc$. However, it feels incorrect, especially if $n$ tends to $\infty$. What I am missing?