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- Real Analysis Notes by Prof Syed Gul Shah
- _n\}$ and $\{t_n\}$ converge to same limits as s, then the sequence $\{u_n\}$ also converges to s. *... rem: If the sequence $\{s_n\}$ converges to //s// then $\exists$ a positive integer such that $\left| {\... {t_n\}$ converge to //s// and //t// respectively. Then (i) $\left\{a{s_n}+b{t_n}\right\}$ converges to $... nce. (i) If $\{s_n\}$ is monotonically increasing then it converges to its supremum. (ii) If $\{s_n\}$ i
- Metric Spaces (Notes)
- sets * Theorem: Let $(X,d)$ be a metric space. Then for any $x,y\in X$, $$\left| {\,d(x,\,A)\, - \,d(... ubset X$. If $x \in X$ is a limit point of //A//. Then every open ball $B(x;r)$ with centre //x// contai... of Sequence * Theorem: If $(x_n)$ is converges then limit of $(x_n)$ is unique. * Theorem: (i) A co... ounded. (ii) ii) If ${x_n}\to x$ and ${y_n}\to y$ then $d(x_n,y_n)\to d(x,y)$. * Cauchy Sequence * T
- Linear Algebra: Important Definitions and Results
- a vector space $V$ has a basis with $n$ elements, then $n$ is called dimension of $V$. * If $W$ is a subspace of $V$, then dim(W) = dim(V). * The span of column of a matr... $A$. * If $T:V\to W$ is linear transformation, then $K(T)$, kernel of $T$, is subspace of $V$. * If $T:V\to W$ is linear transformation, then $R(T)$, range of $T$, is subspace of $W$. {{incl
- Complex Analysis (Quick Review)
- joined end to end that lies entirely in the set, then $S$ is connected. * A function $v(x,y)$ is harm... f $f$ is analytic in simply connected domain $D$, then for every simple closed contour $C$ in $D$, $\int
- Complex Analysis (Easy Notes of Complex Analysis)
- y.org If there is error or mistake in the notes, then please directly contact to Dr. Amir Mahmood for c
- Mechanics (Easy Notes of Mechanics)
- y.org If there is error or mistake in the notes, then please directly contact to Dr. Amir Mahmood for c
- Special Theory of Relativity by M Usman Hamid and M Zeeshan Ahmad
- se in inertial frames of reference. The theory is then called restricted theory or Special Theory of Rel