When I packed my bag to go to Bandung on Wednesday I only put in one shirt for Thursday although I planned to stay until Friday. I thought, “hey, it’s Bandung and I travel there to shop, why would I pack so many shirts? I should just go buy some”. I probably shouldn’t have oversimplified things. Though one of the reason I went to Bandung is to shop, my shopping time frame is pretty tight and sometimes I can be a bit picky. Thus, I fail to find even a single suitable shirt~ when the shirt match, the price doesn’t and vice versa 😆 Luckily I met my junior Oskar who gave me souvenirs from his short visit in Bulgaria for International Mathematics Competition. Why am I so lucky? Because one of the souvenir is T-shirt, yeaaay, problem solved 😉
Thank you so much Os 🙂 Here’s the photo of the souvenirs:
Souvenirs from Bulgaria is not the only thing he gave me; He also gave me a little math problem.
If you can’t see the photo clearly, the problem sounds like this:
Let be a converging sequence of positive real numbers that satisfy the following equation
Proves that is the constant sequence . The problem is really intuitive isn’t it? The answer to this problem is also quite lucid. It took me less than a minute to solve. You may wanna try to tackle the problem yourself before you see my solution below.
Proof: If is the constant sequence then it clearly converges and satisfy the equation. Assume that is not the constant sequence . Then there’s a member of the sequence that is not . Without loss of generality, pick that as and also assume that . Why don’t we lose generality by stating both assumptions?
The equation above can be rewritten as (why?). Since , we have . But then, .
By repeating this step we have .
Since , we know diverge to infinity.