Let {xn} be a sequence of real numbers. Let yn be the average of the first n-terms: X1 + x2 + + Xn Yn n Prove that if {xn} converges to xo, then {yn} also converges to xo. (Hint: using the definition...

Extracted text: Let {xn} be a sequence of real numbers. Let yn be the average of the first n-terms: X1 + x2 + + Xn Yn n Prove that if {xn} converges to xo, then {yn} also converges to xo. (Hint: using the definition of convergence, choose a large threshold N after which |Xn – x∞| is as small as we want, then choose n large enough such that the |x; – x|/n is very small for 1 < i="">< n.)="" consider="" a="" numerical="" series="" 1="" an.="" denote="" by="" sn="of" the="" first="" n-partial="" sums="" be="" lk="1" ak="" the="" partial="" sums.="" let="" the="" average="" s1="" +="" 82="" +="" +="" sn="" on="As" a="" direct="" corollary="" of="" part="" a),="" the="" convergence="" of="" {sn}="" implies="" that="" of="" {on}.="" in="" the="" following="" problems,="" we="" explore="" when="" the="" convergence="" of="" {on}="" implies="" that="" of="" {sn}.="" if="" {on}="" converges="" to="" o,="" and="" ak="o()" as="" k="" →="" 0="" (ak="" is="" “small="" oh"="" of="" ),="" prove="" that="" {sn}="" also="" converges="" to="" o.="" (hint:="" express="" the="" difference="" sn-on="" using="" {kak="" :="" k="1," ·="" ·.="" ,n}.)="" as="" k="" →="" o="" (ak="" is="" “big="" oh"="" of="" if="" {on}="" converges="" to="" oo,="" and="" ak=")," prove="" that="" {sn}="" also="" converges="" to="" ooo="">