Real Analysis Domain Guide

Completeness is the engine. It is why a bounded increasing sequence must converge even when you cannot name the limit, why Cauchy sequences converge, and why $\mathbb{R}$ has no gaps for a limit to fall into. Almost every existence theorem in the domain traces back to it, which is why $\mathbb{Q}$ — where $x_n$ approximating $\sqrt{2}$ is Cauchy but has no limit in the space — is the right counterexample to keep in mind.

Compactness is the hypothesis that upgrades continuity. On a compact domain a continuous function is bounded, attains its extremes, and is uniformly continuous; on a non-compact domain each of those can fail, and the standard failures ($1/x$ on $(0,1)$, $x^2$ on $\mathbb{R}$, $\sin(1/x)$ near $0$) are worth memorising as objects rather than as arguments.

Uniform convergence is about whether the approximation is good everywhere at once. Pointwise convergence lets each $x$ take as long as it likes, and that slack is exactly what allows continuity to be lost in the limit. The supremum test — does $\sup_x |f_n(x) - f(x)|$ go to zero? — converts the question into a computation.

Sequences first. A bounded monotone sequence converges (monotone convergence theorem); every bounded sequence has a convergent subsequence (Bolzano–Weierstrass); and a sequence converges if and only if it is Cauchy. Between them these settle most 'does this converge?' items without any limit computation. For recursively defined sequences the standard exam pattern is: show the sequence is monotone, show it is bounded, conclude convergence, then find the limit by solving the fixed-point equation obtained from letting $n \to \infty$ on both sides — an order that matters, because solving the fixed-point equation first proves nothing about existence.

Series belongs to the same circle of ideas, seen through partial sums. The $n$-th term test is a one-way street: terms failing to approach zero forces divergence, while terms approaching zero says nothing. The integral test converts a series into an improper integral, which is where the $p$-series threshold $p > 1$ comes from. Absolute convergence implies convergence but not conversely, and the difference is not cosmetic: an absolutely convergent series has the same sum under every rearrangement, whereas a conditionally convergent series can be rearranged to converge to any real number at all.

Continuity has three layers on this exam. Continuity at a point is the three-condition definition. Continuity on a set is that condition at every point of it. Uniform continuity demands one $\delta$ for the whole set. The separating examples are compact: $f(x) = 1/x$ is continuous on $(0,1)$ but neither bounded nor uniformly continuous; $f(x) = x^2$ is continuous on $\mathbb{R}$ but not uniformly continuous there; $f(x) = \sqrt{x}$ is uniformly continuous on $[0,\infty)$ but not Lipschitz at $0$; $f(x) = |x|$ is continuous everywhere and differentiable nowhere at $0$. The theorem that ties the layers together is that a continuous function on a compact set is automatically uniformly continuous and attains its maximum and minimum.

The two great value theorems are used constantly and are usually tested by removing a hypothesis. The intermediate value theorem needs continuity on a closed interval and gives every value between $f(a)$ and $f(b)$ — it is the standard route to 'this equation has a root'. The extreme value theorem needs continuity on a closed bounded interval and gives an attained maximum and minimum; drop closedness or boundedness and it fails, which is exactly what $1/x$ on $(0,1)$ demonstrates. The mean value theorem needs continuity on $[a,b]$ and differentiability on $(a,b)$, and it is the engine behind statements like 'if $f' = 0$ throughout an interval then $f$ is constant'.

Sequences of functions is the topic that separates scores. Pointwise convergence is checked one $x$ at a time; uniform convergence is checked by the supremum of the error. Uniform convergence preserves continuity, permits interchange of limit and integral on a bounded interval, and is what makes term-by-term reasoning legitimate. Differentiation is the awkward case: uniform convergence of $f_n$ does not give convergence of $f_n'$, and the standard witness is $f_n(x) = \sin(nx)/n$, which converges uniformly to $0$ while $f_n'(x) = \cos(nx)$ converges nowhere.

Counting and probability are grouped into this domain by the exam's own taxonomy and are answered by a short rule set. Order matters means permutations, $P(n,r) = n!/(n-r)!$; order does not matter means combinations, $C(n,r) = n!/(r!(n-r)!)$. Overcounting is corrected by division, and the classic error is dividing twice for the same symmetry. On the probability side, $P(A \cup B) = P(A) + P(B) - P(A \cap B)$, $P(A \cap B) = P(A \mid B)P(B)$, and independence is the numerical condition $P(A \cap B) = P(A)P(B)$. Expectation is always linear — $E[X + Y] = E[X] + E[Y]$ with no independence needed — while variance adds only for independent variables, and $\operatorname{Var}(aX + b) = a^2 \operatorname{Var}(X)$ shows why the shift disappears and the scale is squared.