Real Analysis
"All The Mathematical Methods I Learned In My University Math Degree Became Obsolete In My Lifetime" - Keith Devlin
Supremem and Infimum of Finite Sets We demonstrate that each finite set has, and contains, a sup and inf. Can you spot the induction? (https://www.desmos.com/calculator/ga339zmvbe)
Algebra of sup and inf Consider the set A={1/2, 2/3, 3/4,...}. We know inf(A)=1/2, sup(A)=1. How does adding or multiplying A by k change these bounds? (https://www.desmos.com/calculator/wgmxgeg2ku)
Triangle Inequality Hit the shuffle button a few times. When is |A+B|=|A|+|B|? When is it less? Greater? (https://www.desmos.com/calculator/d9xmciofp0)
Bounded functions How do the bounds of f and g compare to the bounds of f+g? (https://www.desmos.com/calculator/gswiultpsd)
Convergent sequence Does this sequence appear to be getting closer to some value? Hit shuffle to get a new sequence, click bounds to see the bounds of the sequence. (https://www.desmos.com/calculator/hfmkpnsdsj)
Bolzano–Weierstrass Can we see the monotonic convergent subsequences? Hit shuffle for a new sequence. (https://www.desmos.com/calculator/bfrha7jpvo)
Sequence Continuity My favourite definition of condtiunity: f is continuous if and only if it preserves the limits of sequences! (https://www.desmos.com/calculator/hcrumhavwu)
Sequence disContinuity Discontinuous function do not preserve the limits of sequences. (https://www.desmos.com/calculator/azib9eptvv)
Epsilon-Delta Continuity For each epsilon E, can we find a delta D so that f((a-D, a+d)) is contained in (f(a)-E, f(a)+E)? (https://www.desmos.com/calculator/bzgwtaws41)
Epsilon-Delta dis-Continuity Find an epsilon E, so that for any delta D, f((a-D, a+d)) is never contained in (f(a)-E, f(a)+E) at the point of discontinuity. (https://www.desmos.com/calculator/8l0wod4hbi)
Algebra of sup and inf Consider the set A={1/2, 2/3, 3/4,...}. We know inf(A)=1/2, sup(A)=1. How does adding or multiplying A by k change these bounds? (https://www.desmos.com/calculator/wgmxgeg2ku)
Triangle Inequality Hit the shuffle button a few times. When is |A+B|=|A|+|B|? When is it less? Greater? (https://www.desmos.com/calculator/d9xmciofp0)
Bounded functions How do the bounds of f and g compare to the bounds of f+g? (https://www.desmos.com/calculator/gswiultpsd)
Convergent sequence Does this sequence appear to be getting closer to some value? Hit shuffle to get a new sequence, click bounds to see the bounds of the sequence. (https://www.desmos.com/calculator/hfmkpnsdsj)
Bolzano–Weierstrass Can we see the monotonic convergent subsequences? Hit shuffle for a new sequence. (https://www.desmos.com/calculator/bfrha7jpvo)
Sequence Continuity My favourite definition of condtiunity: f is continuous if and only if it preserves the limits of sequences! (https://www.desmos.com/calculator/hcrumhavwu)
Sequence disContinuity Discontinuous function do not preserve the limits of sequences. (https://www.desmos.com/calculator/azib9eptvv)
Epsilon-Delta Continuity For each epsilon E, can we find a delta D so that f((a-D, a+d)) is contained in (f(a)-E, f(a)+E)? (https://www.desmos.com/calculator/bzgwtaws41)
Epsilon-Delta dis-Continuity Find an epsilon E, so that for any delta D, f((a-D, a+d)) is never contained in (f(a)-E, f(a)+E) at the point of discontinuity. (https://www.desmos.com/calculator/8l0wod4hbi)