To analyze set $E = \{x \mid x = \frac{n^3}{2n^2 + 1}, n \in \mathbb{N} \}$, let’s proceed step by step:

Step 1: Examine the general form of the elements of $E$

Each $x$ in the set $E$ is given by: $x_n = \frac{n^3}{2n^2 + 1} \quad \text{where } n \in \mathbb{N} \, \text{(natural numbers)}.$

Here:

• The numerator is $n^3$,
• The denominator is $2n^2 + 1$.

Step 2: Behavior of $x_n$ as $n \to \infty$

To analyze the behavior of $x_n$, divide the numerator and denominator by $n^2$: $x_n = \frac{n^3}{2n^2 + 1} = \frac{n^3 / n^2}{(2n^2 + 1) / n^2} = \frac{n}{2 + \frac{1}{n^2}}.$

As $n \to \infty$, the $\frac{1}{n^2}$ term in the denominator approaches 0. Thus: $\lim_{n \to \infty} x_n = \frac{n}{2} \to \frac{n}{2} \, (\text{grows without bound}).$

Therefore, the elements of $E$ increase unbounded as $n \to \infty$.

Step 3: Behavior for small values of $n$

To determine the smallest element of the set, substitute small values of $n$ (starting from $n = 1$):

1. For $n = 1$: $x_1 = \frac{1^3}{2 \cdot 1^2 + 1} = \frac{1}{2 + 1} = \frac{1}{3}.$

2. For $n = 2$: $x_2 = \frac{2^3}{2 \cdot 2^2 + 1} = \frac{8}{2 \cdot 4 + 1} = \frac{8}{8 + 1} = \frac{8}{9}.$

3. For $n = 3$: $x_3 = \frac{3^3}{2 \cdot 3^2 + 1} = \frac{27}{2 \cdot 9 + 1} = \frac{27}{18 + 1} = \frac{27}{19}.$

Observe that: $x_1 = \frac{1}{3}, \, x_2 = \frac{8}{9}, \, x_3 = \frac{27}{19}.$

We can see that $x_n$ increases as $n$ increases.

Step 4: Smallest element and behavior of $E$

• The smallest element of $E$ occurs at $n = 1$, where $x_1 = \frac{1}{3}$.
• There is no largest element because $x_n \to \infty$ as $n \to \infty$.

Step 5: Lub (Least Upper Bound) and Glb (Greatest Lower Bound)

• Greatest Lower Bound (glb): The smallest element of $E$ is $\frac{1}{3}$.
• Least Upper Bound (lub): Since $x_n \to \infty$, the set is unbounded above, so there is no finite lub.

Final Answer:

For the set $E$:

1. Smallest element: $\frac{1}{3}$,
2. Largest element: Does not exist (unbounded above),
3. glb (infimum): $\frac{1}{3}$,
4. lub (supremum): $\infty$.

Let me know if you want further explanations or details!

Related Questions:

1. What happens if the denominator of $E$ is modified to $2n^2 - 1$?
2. Can you prove that $x_n$ is strictly increasing for all $n \in \mathbb{N}$?
3. What would the behavior of $x_n$ be if $n$ starts at a different integer (e.g., $n \geq 2$)?
4. How would the sequence behave if $n$ were allowed to take real values?
5. How can you calculate the asymptotic growth rate of $x_n$?

Tip:

To determine bounds of sequences, analyze their behavior as $n \to \infty$ and test small values systematically.