Limits of convergent sequences preserve (non-strict) in- equalities. Theorem 3.22. If (xn) and (yn) are convergent sequences and xn ? yn for all n ? N then.
Let (xn) and (yn) be bounded sequences in R with xnyn ? 0 for all n ? N. (a) Show that lim sup n??. (xnyn) ? (lim sup.
(a) If the sequence (xn) is bounded below and (yn) diverges to +?
(ii) If (xn) and (yn) are convergent sequences then the sequence (xnyn) converges and lim(xnyn) = lim xn limyn. (iii) If (xn) converges to x and x = 0
(b) Prove that if xn ? x and yn ? y as n ? ? then d(xn
subsequence of a convergent sequence is convergent this is impos- a) If (xn) and (yn) are Cauchy sequences
Let {xn} and {yn} be bounded sequences. a) Show that {xn + yn} is bounded. b) Show that. (lim inf n
(b) Every bounded monotonic sequence is a Cauchy sequence. (c) If the sequences (xn) and (yn) diverge then the sequence (xnyn) diverges.
n=1 be Cauchy sequences in a metric space (X d). Then lim n?? d(xn
If (xn) is a null sequence and (yn) is a bounded sequence then thesequence (xnyn) is a null sequence If (xn) and (yn) are convergent sequences then the sequence (xnyn)converges and lim(xnyn) = limxnlimyn If (xn) converges toxandx6= 0 then almost all terms of (xn) arenonzero and the sequence (1/xn) converges to 1/x Proof of (iI)
convergence and divergence bounded sequences continuity and subsequences Relevant theorems such as the Bolzano-Weierstrass theorem will be given and we will apply each concept to a variety of exercises Contents 1 Introduction to Sequences 1 2 Limit of a Sequence 2 3 Divergence and Bounded Sequences 4 4 Continuity 5 5
1 be sequences of real numbers Assume that (x n)1 =1 is bounded and that lim n!1y n= 0 Let (z n) 1 n=1 = (x ny n) 1 =1 Show that lim n!1z n= 0 Hint Exploit the de nitions of convergence boundedness and the properties of the absolute value Possible solution Assume that there exists M 0 such that for all n2N jx nj M and that lim n!1y
Figure 2 4: Sequences bounded above below and both Bounds for Monotonic Sequences Each increasing sequence ( a n) is bounded below by a1 Each decreasing sequence ( a n) is bounded above by a1 Exercise 3 Decide whether each of the sequences de?ned below is bounded above bounded below bounded If it is none of these things then explain why
A sequence (xn) of real numbers is a functionf: N!R wherexn=f(n) We can consider sequences of many di erent types of objects (for examplesequences of functions) but for now we only consider sequences of real numbersand we will refer to them as sequences for short A useful way to visualize asequence (xn) is to plot the graph ofxn 2Rversusn2N
Di?erent sequences of convergent in probability sequences may be combined in much the same way as their real-number counterparts: Theorem 7 4 If X n ?P X and Y n ?P Y and f is continuous then f(X nY n) ?P f(XY) If X = a and Y = b are constant random variables then f only needs to be continuous at (ab)