LeetCode 300 Longest Increasing Subsequence - State Design

Define $dp[i]$ as the length of the longest increasing subsequence ending at index $i$. Why "ending at $i$" instead of "using elements 0 to $i$"? Because when we extend a subsequence, we need to know its last element.

If $dp[i]$ just meant "best in prefix", we wouldn't know what value to compare against. With the "ending at" definition, $arr[i]$ is the last element. We can check if $arr[j] < arr[i]$ for all $j < i$ and extend from there. Base case: $dp[i] = 1$ for all $i$. Every element by itself is a valid increasing subsequence.