Assume X,. X,.... form a sequence of independent RVs, each uni- formly distributed on (0, 1). Let N=min{n :X, + X,+.+ X, > 1}. Show that EN =e.
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- Let X be a set, and define d(x,y) ={0, if x=y; 1, otherwise A sequence (xn) ⊆ X is called eventually constant if there is N ∈ N such that xm=xn; m, n > N. Show that any eventually constant sequence converges.Let {xn} and {yn} be sequences in R with xn → a. Prove that if there exists an N ∈ N such that for all n ≥ N, |xn − yn| ≤ 1/n, then yn → a.If Ean, converges and a, > 0 for all n, can anything be said about E(1/an).?
- Find limits of the following sequences or prove that they are divergent.(a) an =√n(−1)^nLet X be a set, and define d(x,y) ={0, if x=y; 1, otherwise Show that, if a sequence (xn) is convergent under the discrete metric, then (xn) must be eventually constant.Prove using the ϵ−n0 definition that the sequence Xn=(9−7n)/(8−13n) converges, and find its limit.