Prove or disprove: The function is bounded: $f(n) = n + \frac{1}{n}$

Question

Prove or disprove: The function is bounded: $f(n) = n + \frac{1}{n}$ where $n \in \mathbb{R} \setminus 0$

I show that function has infimum and supremum and if it has both it's bounded.

supremum:

$$\lim_{n \rightarrow \infty}\left(n+\frac{1}{n}\right)= \infty$$

infimum:

$$\lim_{n \rightarrow -\infty}\left(n+\frac{1}{n}\right)= -\infty$$

We found that the function has a supremum and an infimum but because both are infinite, function isn't bounded.

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Did I do it correctly? I don't want know another way of doing it (except it doesn't work as I did here), I only want know if you can do it like that and if it's correct please!

Answer 1

Your solution is entirely incorrect.

• You speak of a sequence, but then say that $n\in\mathbb R\setminus \{0\}$. If $a_n$ is a sequence, then $n\in\mathbb N$.
• $2$ is not a supremum, since the third element of the sequence is equal to $3+\frac13$...
• Your calculation of the supremum is not correct. Taking the limit as $n\to 1$ of a sequence (not a function) makes no sense.
• Your calculation of the infimum is not correct. An infimum of a sequence is not the same as the limit of that sequence. And also, the limit, if anything, would be $\infty$.
• You conclude that the sequence has an infimum (which it does), but you "showed" that the infimum is $-\infty$. You cannot conclude that it is bounded from that.