Is this a valid proof of the extreme value theorem? When I research on the internet, I see complicated proofs. My proof seems simple to me. Is it a valid proof?

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i understand why it was invalid. Then is it valid?

 A: Your proof is not valid.
It is not valid for a few reasons. Firstly, you suggest that if $f(x_1) < f(x_2) < \cdots < f(x_n)$, (and so on for arbitrarily long sequences), then necessarily $f(x_n) \to \infty$. This is not true. In fact, the function $f(x) = x$, where $x_i$ is an increasing sequence converging to $1$, is a counterexample.
There is a deeper reason why your proof is not valid, which is that you presuppose that your sequence of $x_i$ converges to some $x$. This brushes under the rug the most challenging aspects of the classical proofs, which concern the topology of the interval, and in particular concern the compactness of the interval.
A: The answer by user "mixedmath" as well as various comments amply describe the problems with your proofs. This is a digression which is too long to fit in a comment. 

I don't know if you are aware of the book Calculus by M. Spivak. The result you are trying to prove here is one of the Three Hard Theorems. The reason Spivak calls these theorems as hard is because these theorems seem very intuitive yet their proofs are not intuitive at all. In fact any approach based on common sense combined with the high school level knowledge of algebra, calculus is not going to get you a proof.
The proofs of these theorems is based on the completeness property of real numbers and once you know about completeness property these results are trivial applications. But unless you are aware of completeness it is just not possible to prove these. Its good that you tried two proofs of extreme value theorem using your intuition and perhaps realized that one of your proofs was invalid. This kind of exercise will at least help you to understand that these results are not actually that intuitive as they appear to be. 
I would suggest you to learn about completeness of real numbers and then you can have a look at the proofs of this theorem in this blog post.
