# Course - Analysis MT22 > Source: https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/ · Updated: 2025-09-30 · Tags: uni, course > The first course in a three part series to serve as an introduction to "real analysis". Real analysis is about putting the idea of things like convergence, limits, continuity, and differentiation on a rigorous footing (in the context of real numbers). > > > This course dealt specifically with results about sequences and series of real numbers, such as the [Bolzano-Weierstrass theorem](https://en.wikipedia.org/wiki/Bolzano%E2%80%93Weierstrass_theorem): if you write down a sequence of real numbers that doesn't shoot off to plus or minus infinity, then you can always find a subsequence of that sequence and a real number such that the subsequence gets infinitely close to that real number. > > > Out of all the maths courses in [Courses MT22](https://ollybritton.com/notes/uni/prelims/mt22/), this felt like the most different from A-level. Since I was still getting into the groove of taking notes at uni, a lot of the notes for this course are organised chronologically in the Lectures section, rather than by concept in the Notes section. - [Course Website](https://courses.maths.ox.ac.uk/course/view.php?id=608) - [Lecture Notes](https://courses.maths.ox.ac.uk/pluginfile.php/25425/mod_resource/content/3/AnalysisI_Lecture_Notes.pdf) - Other analysis courses: - [Course - Analysis II HT23](https://ollybritton.com/notes/uni/prelims/ht23/analysis-ii/) - [Course - Analysis III TT23](https://ollybritton.com/notes/uni/prelims/tt23/analysis-iii/) - ...and in [Part A](https://ollybritton.com/notes/uni/part-a/), the theme continues with: - [Course - Metric Spaces MT23](https://ollybritton.com/notes/uni/part-a/mt23/metric-spaces/) - [Course - Complex Analysis MT23](https://ollybritton.com/notes/uni/part-a/mt23/complex-analysis/) - Other courses this term: [Courses MT22](https://ollybritton.com/notes/uni/prelims/mt22/) ### Notes - [Notes - Analysis I MT22, Rationals and irrationals](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/notes/rationals-and-irrationals/) - [Notes - Analysis I MT22, Infimums and supremums](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/notes/infimums-and-supremums/) - [Notes - Analysis I MT22, Triangle Inequality](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/notes/triangle-inequality/) - [Notes - Analysis I MT22, Approximation property](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/notes/approximation-property/) - [Notes - Analysis I MT22, Algebra of Limits](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/notes/algebra-of-limits/) - [Notes - Analysis I MT22, Limit Inequality Theorem](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/notes/limit-inequality-theorem/) - [Notes - Analysis I MT22, Convergent implies bounded](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/notes/convergent-implies-bounded/) - [Notes - Analysis I MT22, Limits are unique](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/notes/limits-are-unique/) - [Notes - Analysis I MT22, Monotone sequence theorem](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/notes/monotone-sequence-theorem/) - [Notes - Analysis I MT22, Scenic viewpoint theorem](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/notes/scenic-viewpoint-theorem/) - [Notes - Analysis I MT22, Cauchy sequences](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/notes/cauchy-sequences/) - [Notes - Analysis I MT22, Alternating series test](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/notes/alternating-series-test/) - [Notes - Analysis I MT22, Integral test](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/notes/integral-test/) - [Notes - Analysis I MT22, Power series](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/notes/power-series/) - [Notes - Analysis I MT22, Misc](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/notes/misc/) ### Lectures - [Lecture - Analysis MT22, Lecture 1](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/1/) - [Lecture - Analysis MT22, Lecture 2](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/2/) - [Lecture - Analysis MT22, III](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/3/) - [Lecture - Analysis MT22, IV](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/4/) - [Lecture - Analysis MT22, V](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/5/) - [Lecture - Analysis VI](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/6/) - [Lecture - Analysis VII](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/7/) - [Lecture - Analysis MT22, VIII](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/8/) - [Lecture - Analysis MT22, IX](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/9/) - [Lecture - Analysis MT22, X](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/10/) - [Lecture - Analysis MT22, XI](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/11/) - [Lecture - Analysis MT22, XII](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/12/) - [Lecture - Analysis MT22, XIII](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/13/) - [Lecture - Analysis MT22, XIV](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/14/) - [Lecture - Analysis MT22, XV](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/15/) ### Proofs - [Proofs - Analysis I MT22](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/proofs/) ### Problem Sheets - [Sheet 1](https://courses.maths.ox.ac.uk/pluginfile.php/25433/mod_assign/introattachment/0/AnalysisI_Sheet1.pdf?forcedownload=1) - [Sheet 2](https://courses.maths.ox.ac.uk/pluginfile.php/25435/mod_assign/introattachment/0/AnalysisI_Sheet2.pdf?forcedownload=1) - [Sheet 3](https://courses.maths.ox.ac.uk/pluginfile.php/25437/mod_assign/introattachment/0/AnalysisI_Sheet3.pdf?forcedownload=1), [Problem Sheet - Analysis I MT22, III](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/problem-sheets/3/) - [Sheet 4](https://courses.maths.ox.ac.uk/pluginfile.php/25439/mod_assign/introattachment/0/AnalysisI_Sheet4.pdf?forcedownload=1) - [Sheet 5](https://courses.maths.ox.ac.uk/pluginfile.php/25441/mod_assign/introattachment/0/AnalysisI_Sheet5.pdf?forcedownload=1) - [Sheet 6](https://courses.maths.ox.ac.uk/mod/assign/view.php?id=4561&forceview=1), [Problem Sheet - Analysis MT22, VI](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/problem-sheets/6/) - [Sheet 7](https://courses.maths.ox.ac.uk/pluginfile.php/25445/mod_assign/introattachment/0/AnalysisI_Sheet7.pdf?forcedownload=1) --- Olly Britton — https://ollybritton.com. Machine-readable index: https://ollybritton.com/llms.txt