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Suppose [itex]X_1, X_2,\cdots[/itex] be an independent and identically distributed sequence of exponentially distributed random variables with parameter 1.

Now Let [itex]N_n:=\#\{1\leq k\leq n:X_k\geq \log(n)\}[/itex]

I was told that [itex]N_n\xrightarrow{\mathcal{D}}Y[/itex] where [itex]Y\sim [/itex]Poisson(1).

Could anyone give me some directions for this problem? I am totally out of mind.

Thanks.