Lecture 18

15 Outubro 2025, 11:30 • Alexandra Symeonides

Discussion with students of exercises 2. 4. and 6. about mathematical induction.
Exercises 6.8.3, 6.8.5, 6.8.9

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Lecture 17

15 Outubro 2025, 10:00 • Alexandra Symeonides

6. Limits of functions
6.2 Limit of a function at a point (Cauchy definition). Uniqueness of the limit. Ex. 6.8.2 c) f)
Infinite limits at a point and limits at +- infinity. Ex 6.8.2 d) i)
One-sided limits, puncuterd limits and relationship between them Theorems 6.1 and 6.2. Examples

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Lecture 16

13 Outubro 2025, 11:30 • Alexandra Symeonides

Exercises 4.6.8 e)g), 4.6.13 a) b), 4.6.16 a)b)
Mathematical Induction and sum of the first n naturals.

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Lecture 15

8 Outubro 2025, 11:30 • Alexandra Symeonides

Sublimits of sequences that admit a limit, even and odd subsequences.
Upper and lower limits. Limit versus upper and lower limits.
Exercises discussed with the students ex. 4.6.4 iv) 4.6. 5 b) c) e).

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Lecture 14

8 Outubro 2025, 10:00 • Alexandra Symeonides

Comparison theorems for limits.
Subsequences, Bolzano-Weierstrass theorem.
Definition of sub-limits and sub-limits of unbounded sequences.

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