You can prove a sequence converges straight from the epsilon-N definition. You can also argue convergence from monotone and bounded when no formula for the limit is handy.
Given an epsilon, produce an N that works for the sequence 1/n.
Apply the monotone convergence theorem.
Pull a convergent subsequence out of a bounded sequence.
Worked example
Show 1/n converges to 0. Take epsilon = 0.01.
We need 1/n to be under 0.01
That means n has to exceed 100
AnswerTake N = 101, and every n at or beyond N works
Most common mistakeChoosing N before epsilon: a fixed N = 100 fails the moment epsilon drops to 0.001, because N has to depend on epsilon.
You can decide whether an infinite sum converges and separate absolute from conditional convergence. You can also explain why reordering some series changes the total.
Apply the comparison test against a p-series.
Use the alternating series test with the shrinking condition.
Explain how a conditionally convergent series can be rearranged to any sum.
Worked example
Does the alternating sum of 1/n converge? Does it converge absolutely?
Terms alternate and shrink to 0, so it converges to -ln 2
But taking absolute values gives the harmonic series, which diverges
AnswerConverges conditionally, not absolutely
Most common mistakeTreating conditional convergence as if it were absolute: reordering the terms freely can be made to produce any total you like, including infinity.
You can prove continuity with epsilon and delta and use it to guarantee roots and extreme values. You can also tell uniform continuity from plain continuity.
Produce a delta for a given epsilon on a linear function.
Apply the intermediate value theorem to trap a root.
Show 1/x is continuous on (0, 1) but not uniformly continuous there.
Worked example
Show f(x) = 3x is continuous at x = 2, with epsilon = 0.06
The gap in outputs is 3 times the gap in inputs
So we need the input gap under 0.06/3
Answerdelta = 0.02 works
Most common mistakeSetting delta equal to epsilon: delta = 0.06 lets x = 2.05 through, where the output gap is 0.15 and blows past the required 0.06.
You can prove differentiability from the limit definition and apply the mean value theorem. You can also exhibit a function that is continuous but has no derivative.
Use the mean value theorem to bound how much a function can change.
Show the absolute value function has two different one-sided slopes at zero.
Prove that differentiable implies continuous.
Worked example
For f(x) = x squared on [1, 3], find the mean value theorem point c
Average slope = (9 - 1)/(3 - 1) = 4
Set f'(c) = 2c = 4
Answerc = 2, which does lie inside (1, 3)
Most common mistakeRunning the implication backwards: assuming continuity forces differentiability, though the absolute value function is continuous everywhere and has no derivative at 0.
You can define the integral through upper and lower sums and say which functions are integrable. You can also link the integral back to the derivative.
Compute upper and lower sums for a simple partition.
State the criterion that upper minus lower sums can be squeezed as small as you like.
Apply the fundamental theorem to evaluate a definite integral.
Worked example
Lower sum for f(x) = x on [0, 1] with two equal slices
Slices [0, 0.5] and [0.5, 1], with minimums 0 and 0.5
0 x 0.5 + 0.5 x 0.5
AnswerLower sum 0.25, while the true integral is 0.5
Most common mistakeTaking right endpoints for a lower sum: the values 0.5 and 1 give 0.75, which is the upper sum sitting above the true integral, not below it.
You can measure distance in more general settings and talk about open sets, compactness and completeness there. You can also check whether a proposed distance rule is really a metric.
Verify the triangle inequality for a proposed distance rule.
Show a set is open by fitting a small ball inside it around every point.
Explain why the open interval (0, 1) is not complete.
Worked example
Is d(x, y) = (x - y) squared a metric on the real line?
Test the triangle inequality with 0, 1 and 2
d(0, 2) = 4, but d(0, 1) + d(1, 2) = 1 + 1 = 2
Answer4 is bigger than 2, so it is not a metric
Most common mistakeChecking only that distances are non-negative and vanish when the points match, then declaring a metric without ever testing the triangle inequality.
Before timing yourself, check whether you can explain Real numbers from a blank page. Then connect it to Sequences. If either explanation depends on copying a formula, review the unit first and complete two untimed examples.
Use tools to verify, not to choose the method for you
The Calculus calculator can test calculations and representations used in Real Analysis. Make the setup yourself, predict the sign or scale, and compare the tool result with that prediction. Use the formula library to check conditions as well as notation.
Know when to move to the full test
Move from Real Analysis practice to the complete course test after you can correct a missed problem without reopening the worked answer. Record the earliest wrong decision—not only the final score—so the next study session has a precise target.
Before and after the syllabus
Learn the ideas, then practise them
The unit list tells you what is covered. These two pages are where the method is explained and where you find out whether it stuck.
Start with Real numbers if you are following the full sequence. If that unit feels automatic, open the Real Analysis practice page, choose mixed review, and let the first errors identify the earliest prerequisite to revisit.
How do I know I am ready for the course test?
You are ready when you can choose a method without a hint, show the governing steps, and explain why the result is reasonable. Use the complete Real Analysis test only after you can correct practice errors from a blank page.
Which calculator supports this course?
The Calculus calculator supports the calculations and representations used in this course. Use it to test or visualize a result after making your own setup.
