\( \Pr\{Y_n = k\} = \frac {1} {1 - 2^{-n-1}} \frac {1} {2^{k+1}},

k = 0, 1, 2, ..., n \)

Then the cumulative distribution of \( Y_n \),

\( F_{Y_n}(y) = \Pr\{Y_n \leq y\} = \left\{\begin{array}{lll}

0 & \mbox{if} & y < 0 \\

{\displaystyle \frac {1} {1 - 2^{-n-1}} \left(1 -

\frac {1} {2^{\lfloor y\rfloor + 1} } \right) }& \mbox{if} & 0 \leq y < n \\

1 & \mbox{if} & y \geq n

\end{array}\right . \)

\( \lim_{n\to +\infty} F_{Y_n}(y) = \left\{\begin{array}{lll}

0 & \mbox{if} & y < 0 \\

{\displaystyle 1 - \frac {1} {2^{\lfloor y\rfloor + 1} } }& \mbox{if}

& y \geq 0

\end{array}\right .

= F_Y(y) \)