Prove: Suppose that f is a bounded monotone function on aninterval I. Then f has

Question

Prove: Suppose that f is a bounded monotone function on aninterval I. Then f has a limit on the right and a limit on the leftat each interior point of I. Also, f has a one-sided limit at eachendpoint of I. If extends to infinity, then f tends to a limit as xtends to infinity in the appropriate direction. *** PLEASE SHOW ALL WORK Prove: Suppose that f is a bounded monotone function on aninterval I. Then f has a limit on the right and a limit on the leftat each interior point of I. Also, f has a one-sided limit at eachendpoint of I. If extends to infinity, then f tends to a limit as xtends to infinity in the appropriate direction. *** PLEASE SHOW ALL WORK

Explanation / Answer

Without loss of generality, assume f is increasing. Thedecreasing case follows because if f has a left or right limit atx, then so does -f.

Therefore, the sequence f(x_n) satisfies the cauchy criterionand hence convergest to a limit. Call this limit point L. Nowsuppose another sequence y_n converged to x from the left. Then thesequence {z_n} = {x1, y1, x2, y2, x3, y3, ... } would also convergeto x. Then by the argument above, both f(y_n) and f(z_n)converges, but since f(x_n) is a subsequence of f(z_n), lim f(z_n)= lim f(x_n) = L. But since f(y_n) is a subsequence of f(z_n), wemust have that f(y_n) converges to L as well. I have just shownthat for any sequence y_n converging to x from the left, we havef(y_n) converges to L. Hence, f has a limit point from theleft.
The existence of the right limit is entirely similar. Note ingeneral, they won't necessarily be the same limit because f couldmake a discontinous jump at x. Also, the argument for the endpointsis entirely similar as well. The same proof also works for the fextending to infinity case as well.
The existence of the right limit is entirely similar. Note ingeneral, they won't necessarily be the same limit because f couldmake a discontinous jump at x. Also, the argument for the endpointsis entirely similar as well. The same proof also works for the fextending to infinity case as well.
The existence of the right limit is entirely similar. Note ingeneral, they won't necessarily be the same limit because f couldmake a discontinous jump at x. Also, the argument for the endpointsis entirely similar as well. The same proof also works for the fextending to infinity case as well.
The existence of the right limit is entirely similar. Note ingeneral, they won't necessarily be the same limit because f couldmake a discontinous jump at x. Also, the argument for the endpointsis entirely similar as well. The same proof also works for the fextending to infinity case as well.

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