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IDY in < oo and lim - Yn < 0o. Prove that lim,+ 1. Let In > 0. Yn > 0 such that lim,- Yn) < lim,-- In lim,+ Yn: i tn < oo and lim yn < . Prove that lim. In 1. Let In 20, yn 0 such that lim Yn) < limn+In lim + Yr
3. Evaluate the limits x-2 (a) lim 2 (c) lim x-,2x2 +2 ?- (b) lim 2 im 2x +4 (d) Given f(x)x1 -1<x s 2 3-?
real analysis. questions
Prove that if lima In = 0 and > M for some M >0 and in 10 > 0, then lima (ny) - Asume 30 = 2,2-20+ In+1 = In + Prove that this sequence has a limit and find the limit. Prove that lim = L with L < if and only if every subsequence limo n L. Suppose that the sequence {an) is increasing and the sequence {yn) is decreasing. Moreover, lim a n -...
Prove Congruence Property 3
C3 If a=b (mod m) and c < 1, then ac = bc (mod mc)
5. Use Rice's Theorem to prove the undecidablity of the following language. P = {< M > M is a TM and 1011 E L(M)}.
Let U ? Rmxn. Prove that if UTI-In, then n < m.
2) Prove that 1 + 3n < 4n for all n > 1. /5 Marks/
4. Prove that SNS Here r < n and r < m.
Prove the statement is true.
(a) The set A= {(2,y) ERR:22 + y2 <1} is uncountable.
Prove, or give a counter example to disprove the following
statements.
a)
b)
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