Suppose Xn converges weakly to X and each Xn is non-negative. Prove that Suppose Xn converges weakly to X and each Xn has the property that with probability one, (This might arise, for example, if...

Suppose X_n
converges weakly to X and each X_n
is non-negative. Prove that

Suppose X_n
converges weakly to X and each X_n
has the property that with probability one,

(This might arise, for example, if each X_n
is of the form

ds and each Y_n
is bounded by 1.) Prove that X has this same property, that is, with probability one,