$6$

PROBLEM SHEET $1$

Problem $6.[10$ pts$]$ Verify whether the statement below is correct or not. $($T: True; F : False$)[20$pts$]$

$($a$)$ Let $f$:$xY$ be a function. If $\{x_i$subx,iinI$\}$ is a collection of subsets in $x,$ then $f({?}_{\text{i}\text{i}\text{n}\text{I}}{x}_i)={?}_{\text{i}\text{i}\text{n}\text{I}}f(x_i).$

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$($b$)$ Let $f$:$xY$ and $g$:$Y$ be two functions. Then the composition function $g$@$f$ always exists and $f$@$g=g$@$f.$

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$($c$)$ If $A,B$ are sets in $R,$ then sup$(A-B)=$supA$-$supB.

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$($d$)$ The geometric sequence

$1,a,a^2,a^3,$cdots,$a^n,$cdots

satisfies that if $\underset{n}{lim}(x_n-y_n)=\underset{n}{lim}{x}_n-\underset{n}{lim}{y}_n\underset{n}{lim}{x}_n\underset{n}{lim}{y}_{n}0,$ then its $limit$ exists.

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$(e)\underset{n}{lim}(x_n-y_n)=\underset{n}{lim}{x}_n-\underset{n}{lim}{y}_{n}if\underset{n}{lim}{x}_n$ and $\underset{n}{lim}{y}_n$ exist.

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