problem about almost sure convergence

Does \(\displaystyle (X_{n})_{n_{\geq 1}}\), \(\displaystyle X_{n}=\frac{n}{ln(n)}Y_{n}\), where \(\displaystyle Y_{n} \sim Exp(n)\) converge a.s. to 0?

As n goes to zero, for n rational, this converges on 0,
but surely you mean for n natural.
if n>1, the function diverges, as n increases,
if Y = e[sup:152he7pm]-n[/sup:152he7pm], the function converges on zero as n increases.