On a probability space $(\Omega,\mathcal F,\mathbb P)$.
Let $(X_n)_n, (Y_n)_n$ two sequences of random variables.
Question :
If $X_n=Y_n$ for every $n$ and $X_n$ converges $L_1$ to $X$ then $Y_n$ converges to $X$ ?
I think yes since : $\forall \eta>0,\ \exists N>0, \forall n\geq N,\ $ $$ \mathbb E \mid Y_n-X \mid\leq\mathbb E \mid Y_n-X_n \mid+\mathbb E \mid X_n-X \mid \\ =0+\mathbb E \mid X_n-X \mid<\eta $$
However I have a counter example :
If we take $X_n=\epsilon$ (is a Bernouilli$(1/2,\{-1,1\})$ for instance) and $Y_n=X_n 1_{1/n>0}$ then $X_n=Y_n$ but $Y_n$ converges $L_1$ to $0\neq \epsilon$.
Where does my proof (or counter-example) is flawed ?