Let x n , and y n = 1 be two sequences of positive numbers. Assume that x n &lt;&lt; y n Decide whether these claims are always true, always false, or sometimes true and sometimes false (depending on the specific sequences) and prove it.1) x n &lt;&lt; x n + y n 2 2) x n + y n 2 &lt;&lt; y n

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Let             x      n      , and             y      n        =  1 be two sequences of positive numbers. Assume that       x    n    &amp;lt;&amp;lt;      y    n  Decide whether these claims are always true, always false, or sometimes true and sometimes false (depending on the specific sequences) and prove it.1)       x    n    &amp;lt;&amp;lt;                    x        n            +              y        n              2  2)                     x        n            +              y        n              2    &amp;lt;&amp;lt;      y    n

lydalaszq · Accepted Answer

A sequence of real numbers is the function from set of natural numbers to set of real numbers. A sequence of positive numbers means that the range is set of positive numbers.Given:                     x        n                    n      =      1              ∞       and                     y        n                    n      =      1              ∞       of positive numbers, such that       x    n    &amp;lt;&amp;lt;      y    n   for each   n  ∈  N1) Consider       x    n    &amp;lt;&amp;lt;      y    n        x    n    +      x    n    &amp;lt;&amp;lt;      y    n    +      x    n                      x        n            +              x        n              2    &amp;lt;&amp;lt;                    y        n            +              x        n              2        x    n    &amp;lt;&amp;lt;                    y        n            +              x        n              2  Thus, the statement is true for all   n  ∈  N2) Consider      x    n    &amp;lt;&amp;lt;      y    n        x    n    +      y    n    &amp;lt;&amp;lt;      y    n    +      y    n                      x        n            +              y        n              2    &amp;lt;&amp;lt;                    y        n            +              y        n              2                      x        n            +              y        n              2    &amp;lt;&amp;lt;      y    n  Thus, the statement is true for all   n  ∈  N

Holetaug · Accepted Answer

Thanks!

Let <mrow class="MJX-TeXAtom-ORD"> x n </msub> </mrow> , and <mrow clas

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