Question

Let \(f:\mathbb{N}\to\{1,2,3,\ldots,100\}\) be an onto function. Consider the following statements.

• A. \(P\) is correct and \(Q\) is NOT correct
• B. \(P\) is NOT correct and \(Q\) is correct
• C. Both \(P\) and \(Q\) are correct
• D. Neither \(P\) nor \(Q\) is correct

Hint:

The \(\limsup\) and \(\liminf\) of a sequence depend on tail supremum and tail infimum. However, restricting to only even or only odd subsequences can change the limiting value.

Answer 1

The Correct Option is B

Step 1: Read statement $P$.
Here $g(n)=\sup\{f(2n),f(2n+2),\dots\}$ uses only the even indexed tail, while $\limsup f(n)$ uses every term.

Step 2: See why $P$ can fail.
If the big values of $f$ sit on the odd indices, the even tail supremum can miss them. So $\lim g(n)$ need not equal $\limsup f(n)$, and $P$ is not correct.

Step 3: Read statement $Q$.
$h(n)=\inf\{f(n),f(n+1),\dots\}$ is exactly the tail infimum, and by definition $\liminf f(n)=\lim_n h(n)$.

Step 4: Use the subsequence fact.
Since $h(n)$ converges, every subsequence shares the limit. The indices $4n+1$ go to infinity, so $\lim h(4n+1)=\liminf f(n)$. Thus $Q$ is correct.

Step 5: Conclude.
$P$ wrong, $Q$ right, which is option (B).
\[ \boxed{(B)} \]