Big O Proof

🧠 What is Big O really saying?

Big O is a way of asking:

“If I double the amount of work my code has to do… will it take:

• just a little longer?

• exactly twice as long?

• four times as long?

• explode into a black hole of slowness?”

Big O is the science of saying:

"I don’t care how long your code takes on one input.
I care about how much worse it gets as the input size (n) gets bigger and bigger."

🍔 Real-Life Analogy: Fast Food Drive-Thru

Let’s say you work at a burger stand. You want to know:

“If 10 cars show up, how long will it take to serve them?”

Now imagine 100 cars. Does it:

• Take 10× longer? → O(n)

• Take 100×100 longer? → O(n²)

• Take just a tiny bit longer? → O(log n)