:/

docs/math/real-analysis/2-reals.md

@@ -56,7 +56,7 @@
 2. $|ab| = |a||b|$
 3. $|a + b| \leq |a| + |b|$
 
-**Collary**. Givem $a, b \in \mathbb{R}$, then $\abs{\abs{a} - \abs{b}} \leq \abs{a - b}$.
+**Corollary**. Givem $a, b \in \mathbb{R}$, then $\abs{\abs{a} - \abs{b}} \leq \abs{a - b}$.
 
 **Remark**. Every field has at least one absolute value function.
 
@@ -104,4 +104,4 @@ $$
 
 **Theorem**. If $a < b$, then the interval $[a, b]$ is an uncountable set.
 
-**Collary**. $\mathbb{R}$ is uncountable.
+**Corollary**. $\mathbb{R}$ is uncountable.

docs/math/real-analysis/3-sequences-series.md

@@ -96,7 +96,7 @@ is a *subsequence* of $X$,
 
 **Theorem**. Every sequence of real numbers $(x_n)$ contains a monotonic subsequence $(x_{n_k})$.
 
-**Collary**. Bolzano-Weierstrass Theorem. Every bounded sequence of real numbers has a convergent subsequence.
+**Corollary**. Bolzano-Weierstrass Theorem. Every bounded sequence of real numbers has a convergent subsequence.
 
 ## Section 3.5 - The Cauchy Criterion
 
@@ -136,9 +136,9 @@ $$
 \abs{s_m - s_n} = \abs{\sum_{i = m + 1}^n x_i} < \epsilon
 $$
 
-**Collary**. $n$-th Term Test. If $\sum x_n$ converges, then $x_n \rightarrow 0$.
+**Corollary**. $n$-th Term Test. If $\sum x_n$ converges, then $x_n \rightarrow 0$.
 
-**Collary**. Absolute Convergence Test. If $\sum \abs{x_n}$ converges, then $\sum x_n$ converges.
+**Corollary**. Absolute Convergence Test. If $\sum \abs{x_n}$ converges, then $\sum x_n$ converges.
 
 ---
 

docs/math/real-analysis/4-limits.md

@@ -6,11 +6,11 @@
 
 **Theorem**. A real number $c$ is a cluster point for a set $A$ if and only if there exists a sequence $(a_n)$ in $A\\ \{c\}$ such that $a_n \rightarrow c$
 
-**Collary**. A real number $c$ is a cluster point of a set $A$ if and only if every $\delta$-neighborhood conttains infinitely many points of $A$.
+**Corollary**. A real number $c$ is a cluster point of a set $A$ if and only if every $\delta$-neighborhood conttains infinitely many points of $A$.
 
 **Definition**. The set of every cluster point of $A$ is called the *derived set* of $A$, and denoted $A'$.
 
-**Collary**. A set $A$ is closed if and only if $A' \subseteq A$.
+**Corollary**. A set $A$ is closed if and only if $A' \subseteq A$.
 
 **Remark**. If $A'$ is the derived set of $A$, then $A'' \subseteq A'$.
 
@@ -58,7 +58,7 @@ $$
 \lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \frac{L}{M}
 $$
 
-**Collary**. If $p, q \in \mathbb{R}[x]$, and $q(c) \neq 0$ for some $c \in \mathbb{R}$, then
+**Corollary**. If $p, q \in \mathbb{R}[x]$, and $q(c) \neq 0$ for some $c \in \mathbb{R}$, then
 
 $$
 \lim_{x \rightarrow c} p(x) = p(c)

docs/math/real-analysis/5-continuity.md

@@ -43,21 +43,21 @@ $$
 \lim_{x \rightarrow c} g(f(x)) = g(L) = g(\lim_{x \rightarrow c} f(x))
 $$
 
-**Collary**. let $A, B \subseteq  \mathbb{R}$, with $f: A \rightarrow B$ and $g: B \rightarrow \mathbb{R}$. If $f$ is continuous at $a \in A$ and $g$ is continuous at $f(a) \in B$, then $g(f(x))$ is continuous at $a$.
+**Corollary**. let $A, B \subseteq  \mathbb{R}$, with $f: A \rightarrow B$ and $g: B \rightarrow \mathbb{R}$. If $f$ is continuous at $a \in A$ and $g$ is continuous at $f(a) \in B$, then $g(f(x))$ is continuous at $a$.
 
 ## Section 5.3 - continuous functions on Intervals
 
 **Theorem**. Let $S, T$ be metric spaces with $A \subseteq S$ and $f: A \rightarrow T$. If $A$ is a compact subset of $S$, then $f(A)$ is a compact subset of $T$.
 
-**Collary**. Let $f: A \rightarrow \mathbb{R}$ be a continuous function, with $A$ being a compact subset of metric space $S$. Then, $f(A)$ is closed and bounded. Moreover, there exists a $p, q \in A$ such that $f(p)$ and $f(q)$ are the supremum and infimum of $f(A)$.
+**Corollary**. Let $f: A \rightarrow \mathbb{R}$ be a continuous function, with $A$ being a compact subset of metric space $S$. Then, $f(A)$ is closed and bounded. Moreover, there exists a $p, q \in A$ such that $f(p)$ and $f(q)$ are the supremum and infimum of $f(A)$.
 
-**Collary**. Maximum-Minimum Theorem. If $I = [a, b]$ is a closed and bounded interval and $f: I \rightarrow \mathbb{R}$ is continuous on $I$, then $f$ has an absolute minumum and maximum on $I$.
+**Corollary**. Maximum-Minimum Theorem. If $I = [a, b]$ is a closed and bounded interval and $f: I \rightarrow \mathbb{R}$ is continuous on $I$, then $f$ has an absolute minumum and maximum on $I$.
 
 ---
 
 **Theorem**. Let $S, T$ be metric spaces and $A \subseteq S$. Then, if $f: A \rightarrow T$ is continuous on $A$, and $A$ is a connected subset of $S$, then $f(A)$ is a connected subset of $T$.
 
-**Collary**. Suppose that $I$ is an interval. Let $f: I \rightarrow \mathbb{R}$ be continuous on $I$. Then, $f(I)$ is an intterval.
+**Corollary**. Suppose that $I$ is an interval. Let $f: I \rightarrow \mathbb{R}$ be continuous on $I$. Then, $f(I)$ is an intterval.
 
 **Theorem**. (Bolzano's) Invermediate Value Theorem. Suppose $f: [a, b] \rightarrow \mathbb{R}$ is continuous on $[a, b]$ with $a \neq b$. Then, given some $k$ such that $f(a) < k < f(b)$, there exists some $c \in (a, b)$ such that $k = f(c)$.
 
@@ -75,6 +75,6 @@ Note that if $f$ is uniformly continuous, it must be continuous on $A$.
 
 **Remark**. If $S, T$ are metric spaces, $K$ is a compact subset of $S$, and $f: K \rightarrow T$ is continuous on $K$, then $f$ is uniformly continuous.
 
-**Theorem**. Suppose $A \subseteq \mathbb{R}$ and $f: A \rightaarrow \mathbb{R}$ is uniformly continuous. Then, if $(x_n)$ is a Cauchy sequence in $A$, $(f(x_n))$ is a Cauchy sequence in $\mathbb{R}$.
+**Theorem**. Suppose $A \subseteq \mathbb{R}$ and $f: A \rightarrow \mathbb{R}$ is uniformly continuous. Then, if $(x_n)$ is a Cauchy sequence in $A$, $(f(x_n))$ is a Cauchy sequence in $\mathbb{R}$.
 
-**Remark**. Suppose $S, T$ are metric spaces and $f: S \rightaarrow T$ is uniformly continuous. Then, if $(x_n)$ is a Cauchy sequence in $S$, $(f(x_n))$ is a Cauchy sequence in $T$.
+**Remark**. Suppose $S, T$ are metric spaces and $f: S \rightarrow T$ is uniformly continuous. Then, if $(x_n)$ is a Cauchy sequence in $S$, $(f(x_n))$ is a Cauchy sequence in $T$.