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Only step you're missing is to show that the limit approaches e from below for positive numbers (which it does). If the limit approached e from above instead, your solution would not be valid without more steps.
I was going to generalize from a lower number but I found out when plotting x^x versus (x-1)^(x+1) that the inequality flips between x=4.1410415253107 and x=4.1410415254108. In simpler terms, with easier numbers, 4^4 > 3^5 but 5^5 < 4^6.
Only step you're missing is to show that the limit approaches e from below for positive numbers (which it does). If the limit approached e from above instead, your solution would not be valid without more steps.
I was going to generalize from a lower number but I found out when plotting x^x versus (x-1)^(x+1) that the inequality flips between x=4.1410415253107 and x=4.1410415254108.
In simpler terms, with easier numbers, 4^4 > 3^5 but 5^5 < 4^6.
Just change to:
(2023+1)^2024 2023*2023^2024
And devide both by 2023^2024.
No need for limits or anything
As others here have mentioned, this is not a complete proof. You have not shown that this specific approximation of e is < 3.