Define a sequence (fn) from n=1 to infinity of functions

[henry22]

Homework Statement

Define a sequence (fn) from n=1 to infinity of functions on [0,1] by fn(t)=t^n
does the sequence converge in (CL^2[0,1],||.||2)

Homework Equations

The Attempt at a Solution

I am struggling on where to start. I am fairly new to the L2 space and so would just like a bit of help starting.

 

[HallsofIvy]

The L2 norm of function f(t) in this space is $$\sqrt{\int_0^1 f^2(t)dt$$.

Here, that means that if this sequence to a function, f, then we must have that, given any $\epsilon> 0$,
$$||f- t^n||= \sqrt{\int_0^1 (f(t)- t^n)^2 dt}< \epsilon$$
for sufficiently large n (n> N for some integer N).

But just to determine whether or not it converges it might be better to show determine whether or not this is a Cauchy sequence. That is, determine if
$$||t^n- t^m||= \sqrt{\int_0^1 (t^n- t^m)^2 dt}= \sqrt{\int_0^1(t^{2n}- 2t^{n+m}+ t^{2m})dt}$$
goes to 0 as both m and n go to infinity.

(Since $\sqrt{a_n}$ goes to 0 if and only if $a_n$ goes to 0, you can ignore the square root.)