Difference between Increasing sequence of functions and Sequence of increasing functions Suppose we define a sequence of functions $\{f_n \}_{n \in \mathbb{N}}$. I am confused in the following terminologies regarding this sequence of functions - 
1) Increasing sequence of functions
2) Sequence of increasing functions
I suppose these are two different notions but I am unable to distinguish between them. This question can be formulated for sequence of numbers also.
 A: It's like the difference between a growing mob of people and a mob of growing people. 


*

*The sequence is increasing if $f_{m}(x) > f_n(x)$ for $m > n$, for every $x$. 

*The functions are increasing if $f_n(y) > f_n(x)$  for $y > x$, for every $n$.
A: Examples may help:
Let $f_n(x)=n+\sin x$ and $g_n(x)=e^{nx}$.
Then $\{f_n\}_{n\in\Bbb N}$ is a sequence of functions and this sequence is increasing because for $n_1>n_2$ we have $f_{n_1}>f_{n_2}$ (which is just short for $\forall x\in\Bbb R\colon f_{n_1}(x)>f_{n_2}(x)$. However, the individual functions $f_n$ are oscillating, not increasing.
On the other hand,  $\{g_n\}_{n\in\Bbb N}$ is another sequence of functions and this sequence is a sequence of increasing functions, which just say that each $g_n$ individually is increasing, i.e., whenever $x_1>x_2$ then $g_n(x_1)>g_n(x_2)$. But the sequence itself is neither increasing nor decreasing: If $n_1>n_2$ then $g_{n_1}(1)>g_{n_2}(x)$, but $g_{n_1}(-1)<g_{n_2}(x)$, i.e., $g_{n_1}$ and $g_{n_2}$ are not even comparable.

"This question can be formulated for sequence of numbers also." - Actually it can't, or at least we are facing a linguistic rather than a mathematical problem. 
A "sequence of increasing numbers" (as oppsed to an incrasing sequence of numbers) should be a sequence of numbers where the individual numbers are increasing. But a single number cannot be increasing (whereas a function can!). Admittedly, this is not the full truth: We would immediately understand that "a sequence of distinct numbers" is a sequence $(a_n)_{n\in\Bbb N}$ where $a_n\ne a_m$ whenever $n\ne m$, even though being distinct is not a property of an individual number. This time, we cannot mend the situation by speaking of "a distinct sequence of numbers" because that would just be another sequence that differs from a previous sequence, whereas all other properties of the sequence are left in the dark. One might try to formulate "a sequence of pairwise distinct numbers", but let's face it: If we say "a sequence of distinct numbers", then the audience will understand it as intended. The same goes for "a sequence of increasing numbers", as the literal interpretation is not possible. 
However, whenever we consider sequences (or other "super-objects") of objects and refer to a property that could be applied both to the sequence and to the individual objects, then it is a must to be precise in formulation and to differentiate between "a foobar super-object of objects" and "a super-object of foobar objects" (and even "a foobar super-object of foobar objects"). As it is always preferable to make precise formulations in math (well, maybe not always, as I like to say: "Generalizations are always wrong"), I stringly recommend to avoid those formulations I described as understandable and suggest to speak only of an "increasing sequence of numbers" or "a sequence of pairwise distinct numbers".
