TITLE: Convergence in distribution example
QUESTION [0 upvotes]: I have a trouble understanding how below is true:

I have drawn $F_{\frac{Y_n}{n}} (y)$ above. Is this correct?
Now, if I send n to infinity, I still get the same graph (discrete) and not the distribution function of continuous uniform [0,1]?

REPLY [1 votes]: I have drawn $F_{\frac{Y_n}{n}} (y)$ above. Is this correct?

No, the CDF of $Y_n/n$ is a stairway from altitude $0$ to altitude $1$ made of $n$ steps of height $1/n$ from each $(k-1)/n$ to each $k/n$ for $1\leqslant k\leqslant n$. 
Hence the pointwise convergence of $F_{Y_n/n}$ to the CDF $F:x\mapsto\max(\min(x,1),0)$ of the uniform distribution on $(0,1)$.
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