1. Circle the first error and explain what is incorrect about it. If the proof is not correct, can it be fixed to prove the claim true? If it is correct, then move on to the next problem. [HINT: Look at each sentence very carefully] Claim: If f : [a, b] → R is continuous and not constant, then its range is a closed, bounded interval. Proof: Since f is continuous on a compact domain, its range is compact and hence closed and bounded. Since the range is bounded, it has a supremum and an infimum. Since the range is also closed, the supremum and infimum belong to the range. Therefore the range is the closed and bounded interval [m, M], where m is the infimum and M is the supremum of the range.

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.