-- easy proof with probability statements.

Dear forumers,

I am working on the following problem.
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If Yn is bounded in probability, meaning P(|Yn| =< K) > 1 -delta for
all n > n0, and if
Kn is any sequence of constants tending to infty, then P(|Yn| < Kn)
goes to 1 as n goes to infty
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By hypothesis
P(|Yn| =< K) > 1 - delta for all n > n0
For every epsilon, there exists n1 such that for all n > n1, |Kn|
epsilon

Take epsilon=K, there exists n2 such that for all n > n2, |Kn|>K

|Yn| =< K for n>n1 => |Yn| < Kn for n > max(n1,n2)

1-delta <P(|Yn| =< K) < P(|Yn| < Kn) for n > max(n1,n2)
now take delta to 0

Is this rigorous enough? How can I improve my argument.

I greatly appreciate your help.
.

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