To prove the comparison principle for sequences $(a_n)_{n=1}^N$ and $(b_n)_{n=1}^N$ such that $a_n \leq b_n$ for all $n$, we want to show that:

$\sup(a_n)_{n=1}^N \leq \sup(b_n)_{n=1}^N.$

Definitions:

• The supremum $\sup(a_n)_{n=1}^N$ is the least upper bound of the set $\{a_1, a_2, \ldots, a_N\}$, i.e., it is the smallest real number $\alpha$ such that $a_n \leq \alpha$ for all $n$.
• Similarly, $\sup(b_n)_{n=1}^N$ is the least upper bound of the set $\{b_1, b_2, \ldots, b_N\}$, i.e., the smallest real number $\beta$ such that $b_n \leq \beta$ for all $n$.

Step-by-Step Proof:

1. Given: $a_n \leq b_n$ for all $n \in \{1, 2, \ldots, N\}$.

2. Let $\sup(a_n) = \alpha$ and $\sup(b_n) = \beta$. By the definition of supremum, we know that:

   • $a_n \leq \alpha$ for all $n$, and $\alpha$ is the smallest number with this property.
   • $b_n \leq \beta$ for all $n$, and $\beta$ is the smallest number with this property.

3. Upper bound property: Since $a_n \leq b_n$ for all $n$, and $\beta$ is an upper bound for the sequence $(b_n)_{n=1}^N$, it follows that $\beta$ is also an upper bound for the sequence $(a_n)_{n=1}^N$, because for each $n$, $a_n \leq b_n \leq \beta.$ Therefore, $\beta$ is an upper bound for the set $\{a_1, a_2, \ldots, a_N\}$.

4. Smallest upper bound: Since $\alpha = \sup(a_n)$ is the least upper bound of $\{a_1, a_2, \ldots, a_N\}$, and $\beta$ is an upper bound for this set, we must have $\alpha \leq \beta$.

   In other words, $\sup(a_n)_{n=1}^N \leq \sup(b_n)_{n=1}^N.$

Conclusion:

Thus, the comparison principle holds, and we have proven that if $a_n \leq b_n$ for all $n$, then $\sup(a_n)_{n=1}^N \leq \sup(b_n)_{n=1}^N.$

Would you like more details or further clarification?

Here are some related questions to expand on this topic:

1. How does the infimum behave under similar conditions $a_n \leq b_n$ for all $n$?
2. What changes in the comparison principle when the sequences are infinite?
3. Can you prove the comparison principle for integrals in a similar fashion?
4. How does the comparison principle apply in the context of limits of sequences?
5. Can you give an example of two sequences $(a_n)$ and $(b_n)$ to illustrate this principle?

Tip: When dealing with supremum or infimum, always consider whether the sequence is bounded and whether the bounds are attained.