Real Analysis Definitions

Insert just definitions here, not proofs.
Convergent Sequence/Limit of a Sequence: the limit of a sequence is the value that the terms of the sequence ‘tend to’. A sequence is convergent if such a limit exists. Often denoted as x_n \to x or \lim_{n \to \infty} x_n = x.
* In \reals: (x_{n}) converges to x—or equivalently, x is the limit of(x_{n})—if \forall \varepsilon > 0, there exists an N \in \natnums such that \forall n > N, we have: |x - x_{n}| < \varepsilon.
* In metric spaces: (x_{n}) converges to a point in the metric space (X, d), x \in X—or equivalently, x \in X is the limit of(x_{n})—if \forall \varepsilon > 0, there exists an N \in \natnums such that \forall n > N, we have: d(x - x_{n}) < \varepsilon.
* In topological spaces: (x_{n}) converges to a point in the topological space (X, \tau), x \in X—or equivalently, x \in X is the limit of(x_{n})—if for every neighborhood Uof x, there exists an N \in \natnums such that \forall n > N, we have: x_{n} \in U.
* Inequality Rule for Real Sequences: if (x_n)_n, (y_n)_n \subseteq \mathbb{R} and x_n \geq y_n for all n \geq N, then \lim_{n \to \infty} x_n \geq \lim_{n \to \infty} y_n. In other words, if x_n \to x, ~y_n \to y and x_n \geq y_n ~\forall n \geq N \implies x \geq y.
*Infimum: greatest lower bound. If it exists, it is unique.
* Supremum: least upper bound. If it exists, it is unique.
* Set: a collection of different/distinct things (elements). Note: sets do not have any concept of order, nor do repeated elements alter the set: \{1, 2\} = \{2, 1\} = \{1, 1, 2\}. The set with no elements in it is the empty set, \varnothing = \{\}, which is a subset of every set. 
*Open set: 
*Closed set: S is closed if its complement S^{c} is open. Equivalently, if it contains the limit of any convergent sequence in the set. (A bounded set need not contain its boundaries; if it contains none of its boundaries it is open, if it contains all its boundaries, it is closed.)
*Bounded set: a subset S of a metric space is bounded if \exists r > 0 such that \forall x, y \in S, we have d(x, y) < r. In the real numbers, a set is bounded if it has upper bound and lower bounds, therefore a set of real numbers is bounded if it is contained in a finite interval.
*Compact set: a set S is compact if and only if it is sequentially compact, or a.k.a. every open cover of S has a finite subcover. (Note: the Heine-Borel Theorem shows that in Euclidean spaces, S is closed and bounded \iffS is compact). 
* Closure of a set: the closure of a set S (denoted \bar{S}) is the intersection of all closed sets containing S. Informally, this is the smallest closed set that contains S. Also, \bar{S} = S \cup \{ \lim_{n \to \infty} a_n : a_n \in A \forall n \in \natnums \}
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*Dense set: S \subseteq X is dense in X if \forall x \in X, x \in S or x is ‘arbitrarily close’ to an element in S.
*Order-dense set: for any two elements (that have an order defined between them), there is an additional element in between them. A partial or total order, R, on a set X is dense if \forall x, y \in X for which xRy, there \exists z \in X such that xRzRy.
*Power set: the ‘set’ (class/family) of all possible subsets of a set, denoted by 2^{S} := \{X|X \subseteq S\}.
Functions
* Continuous function: Let (X, d) and (X', d') be metric spaces. A function f: X \to X' is continuous at the point x_{0} if for any \varepsilon > 0, there exists a \delta > 0 s.t. d(x, x_{0}) < \delta implies d'(f(x),f(x_{0})) < \varepsilon whenever x \in X . A function is continuous if it is continuous at every point x_{0} \in X.
*Graph of a function: the graph of a function f: X \to Y, is the set Gr f := \{ (x, f(x)) | x \in X\} = \{ (x, y) \in X \times Y | y = f(x) \}.
Correspondences
* Graph of a correspondence: the graph of a correspondence F: X \rightrightarrows Y is the set Gr F := \{ (x, y) \in X \times Y | y \in F(x) \}. (note that the first definition in the graph of a function no longer works because f(x) is a set, so that the ordered pair would have a set as the second object).
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=== Quasiconvex/quasiconcave ===
A function f: X \to \realsdefined on a convex subset S of a real vector space is quasiconvex if for all x, y \in S we have:
* Convex function: f(\lambda x + (1 - \lambda)y) \leq \lambda f(x) + (1 - \lambda) f(y), where \lambda \in [0,1]
*Concave function:f(\lambda x + (1 - \lambda)y) \geq \lambda f(x) + (1 - \lambda) f(y), where \lambda \in [0,1]
*Strictly convex function:f(\lambda x + (1 - \lambda)y) < \lambda f(x) + (1 - \lambda) f(y), where \lambda \in (0,1)
*Strictly concave function:f(\lambda x + (1 - \lambda)y) > \lambda f(x) + (1 - \lambda) f(y), where \lambda \in (0,1)
*Quasiconvex function: f(\lambda x + (1 - \lambda)y) \leq max\{f(x), f(y)\}, where \lambda \in [0,1]
* Quasiconcave function: f(\lambda x + (1 - \lambda)y) \geq min\{f(x), f(y)\}, where \lambda \in [0,1]
* Strictly quasiconvex function: f(\lambda x + (1 - \lambda)y) < max\{f(x), f(y)\}, where \lambda \in (0,1)
* Strictly quasiconcave function: f(\lambda x + (1 - \lambda)y) > min\{f(x), f(y)\}, where \lambda \in (0,1)
A function  defined on a convex subset  of a real vector space is quasiconvex if for all  and  we have
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