Dynamic Programming Challenge: Longest Increasing Subsequence

Given an array of integers, find the length of the longest increasing subsequence. A subsequence is a sequence that can be derived from another sequence by deleting some or no elements without changing the order of the remaining elements.

For example:

#include <iostream>
#include <vector>

int longestIncreasingSubsequence(const std::vector<int>& nums) {
    // Your dynamic programming solution goes here.

    // Return the length of the longest increasing subsequence.
}

int main() {
    std::vector<int> sequence = {10, 22, 9, 33, 21, 50, 41, 60, 80};
    int result = longestIncreasingSubsequence(sequence);

    std::cout << "Length of the longest increasing subsequence: " << result << std::endl;

    return 0;
}

I'm specifically interested in implementing this using dynamic programming techniques. How can I approach this problem using dynamic programming, and what would be the C++ code for solving it? Any insights, code snippets, or explanations would be incredibly helpful in mastering dynamic programming for this particular challenge. Thank you for your assistance!