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: Understand statement \(P\).
Here,
\[ g(n)=\sup\{f(2n),f(2n+2),f(2n+4),\ldots\} \]
This means \(g(n)\) considers only the tail values of the even-indexed subsequence of \(f\).
So,
\[ \lim_{n\to\infty}g(n) \] represents the limiting supremum of the even-indexed terms only.
But
\[ \limsup_{n\to\infty}f(n) \] considers all terms of the sequence, including both even-indexed and odd-indexed terms.

Step 2: Check whether statement \(P\) is always true.
Statement \(P\) need not be true because the largest limiting values of \(f(n)\) may occur along the odd-indexed subsequence.
For example, after finitely many terms, suppose the even-indexed values remain small and the odd-indexed values remain large.
Then the supremum of the even tail can be different from the limsup of the entire sequence.
Hence,
\[ \lim_{n\to\infty}g(n) \neq \limsup_{n\to\infty}f(n) \] in general.
Therefore, statement \(P\) is NOT correct.

Step 3: Understand statement \(Q\).
Here,
\[ h(n)=\inf\{f(n),f(n+1),f(n+2),\ldots\} \]
This is the infimum of the tail of the sequence starting from \(n\).
By definition,
\[ \liminf_{n\to\infty}f(n) = \lim_{n\to\infty}\inf\{f(n),f(n+1),f(n+2),\ldots\} \]
Thus,
\[ \liminf_{n\to\infty}f(n)=\lim_{n\to\infty}h(n) \]

Step 4: Analyze \(\lim h(4n+1)\).
Since \(4n+1\to\infty\) as \(n\to\infty\), the sequence
\[ h(4n+1) \] is a subsequence of
\[ h(n) \]
Because \(h(n)\) has a limit equal to \(\liminf f(n)\), every subsequence of \(h(n)\) has the same limit.
Therefore,
\[ \lim_{n\to\infty}h(4n+1) = \lim_{n\to\infty}h(n) = \liminf_{n\to\infty}f(n) \]
Hence, statement \(Q\) is correct.

Step 5: Final conclusion.
Statement \(P\) is not correct because it considers only even-indexed terms.
Statement \(Q\) is correct because \(h(n)\) is the tail-infimum sequence used in the definition of \(\liminf\).
Therefore, the correct option is
\[ \boxed{(B)} \]