We are given the sequence $s_n = x^{1/n}$, where $0 < x < 1$, and the task is to prove that this sequence converges and find its limit. We're also instructed to use the Monotone Convergence Theorem and consider the subsequence $s_{2n}$.

Step 1: Intuition about the limit

The sequence is $s_n = x^{1/n}$, where $x$ is a number between 0 and 1. As $n \to \infty$, the exponent $\frac{1}{n} \to 0$, which implies:

$s_n = x^{1/n} \to x^0 = 1 \quad \text{as} \quad n \to \infty.$

Thus, intuitively, the sequence $s_n$ seems to converge to 1.

Step 2: Monotonicity of the sequence

Let's analyze whether the sequence is monotonic. To do this, we can check if the sequence is decreasing or increasing by comparing $s_n$ and $s_{n+1}$.

$s_{n+1} = x^{1/(n+1)}.$

Now, we want to check if $s_{n+1} \leq s_n$, i.e., if $x^{1/(n+1)} \leq x^{1/n}$. This is equivalent to:

$\left( x^{1/n} \right)^n \leq \left( x^{1/(n+1)} \right)^{n+1}.$

Since $0 < x < 1$, the function $x^{1/n}$ is increasing as $n \to \infty$, which implies that the sequence is increasing and bounded above by 1. By the Monotone Convergence Theorem, the sequence converges.

Step 3: Subsequence analysis

The hint also suggests considering the subsequence $s_{2n} = x^{1/(2n)}$. Similar to the main sequence, as $n \to \infty$, $\frac{1}{2n} \to 0$, so $s_{2n} \to x^0 = 1$.

Step 4: Conclusion

The sequence $s_n = x^{1/n}$ is increasing and bounded above by 1, which means it converges by the Monotone Convergence Theorem. Therefore, the limit of the sequence is:

$\lim_{n \to \infty} s_n = 1.$

Would you like more details on any part of the solution?

Here are 5 follow-up questions to explore further:

1. How does the Monotone Convergence Theorem guarantee convergence in sequences?
2. What happens to the sequence if $x > 1$ instead of $0 < x < 1$?
3. How would the behavior of the sequence change if $x = 1$?
4. Could you extend this analysis to more complex sequences, like $x^{1/n^k}$ for some integer $k$?
5. How could we apply similar reasoning to sequences involving logarithms or trigonometric functions?

Tip: Monotonicity is a powerful property that simplifies proving convergence when you can establish a sequence is bounded and either increasing or decreasing.