I'm suggesting these because they're "tabletop" telescopes, meaning you just have to put them on a flat, level surface: it comes with its own "mount" so that you don't have to spend more money buying an equatorial mount (although there's nothing wrong with that and it's perfectly possible). There are two instruments that I can suggest for a small child. However, if you insist on buying a telescope, I suggest that you start out with something relatively inexpensive. If you buy the binoculars and a good guide, it'll be better than spending your budget on the telescope (and no guide).
When you don't know the night sky (constellations, etc), finding something on the sky without knowing your way arround is a good and fast way to frustration (and throwing the telescope into the attic).There is no such problem with binoculars since it has image-correcting prisms (it takes up a substancial amount of the binoculars' weight). Many first-timers become confused when they pick a telescope because the image on the ocular is inverted they push the telescope to the right hoping to see something "on the right" and instead the image goes left, they push the telescope up and the image "goes down".A binoculars makes it easier for them because they don't have to make extra effort. It's usual for small children to have trouble closing one eye and looking through a telescope's ocular (even though there are attachments which split the light, thus costing more).It can also find application when you go out camping for example, because you can carry it in the back pack to watch animal life.
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On the other hand, a telescope "funnels" the available light from the object itself (which might be feeble) the amplification of the image comes second, in that respect, so if you amplify an image you're spreading the same light over a wider area. It is no such thing a microscope's purpose is to magnify the image of a small object, and it uses a "strong" backlight for illuminating the object under the eyepiece. I get the impression that most people unfamiliar with telescopes think of a telescope as a "microscope pointing upwards". Telescopes (and binoculars) are, primarily, light-concentrating devices.